{"id":21,"date":"2019-06-10T13:24:00","date_gmt":"2019-06-10T17:24:00","guid":{"rendered":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/chapter\/unit-6-polynomials\/"},"modified":"2024-01-05T09:32:23","modified_gmt":"2024-01-05T14:32:23","slug":"unit-6-polynomials","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/pohmath\/chapter\/unit-6-polynomials\/","title":{"raw":"1.1. Introduction to Algebra","rendered":"1.1. Introduction to Algebra"},"content":{"raw":"
\r\n\r\n[Latexpage]\r\n

Introduction to Algebra<\/h1>\r\n<\/div>\r\nReview of basic algebraic terms:<\/strong>\r\n\r\n\r\n\r\n
Algebraic term<\/strong><\/td>\r\nDescription<\/strong><\/td>\r\nExample<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u00a0Algebraic expression<\/strong><\/td>\r\nA mathematical phrase that contains numbers, variables (letters), and arithmetic operations (+, \u2013<\/em> , \u00d7, \u00f7, etc.).<\/td>\r\n\r\n

3x<\/em> \u2013 4<\/p>\r\n

5a<\/em>2<\/sup> \u2013 <\/em>b + <\/em>3<\/p>\r\n

12y<\/em>3<\/sup> + 7y<\/em>2<\/sup> \u2013<\/em> 5y <\/em>+ $\\frac{2}{3}$<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Constant<\/strong><\/td>\r\nA number on its own.<\/td>\r\n2y + 5\u00a0 \u00a0 \u00a0 \u00a0 constant:\u00a0 \u00a0 \u00a0 \u00a0 5<\/td>\r\n<\/tr>\r\n
Coefficient<\/strong><\/td>\r\nThe number in front of a variable.<\/td>\r\n\r\n

-9x2<\/sup>\u00a0 \u00a0 <\/em>coefficient:\u00a0 \u00a0 -9<\/p>\r\n

\u00a0x<\/em>\u00a0 \u00a0 coefficient:\u00a0 \u00a0\u00a01<\/p>\r\n

(x<\/em> = 1 \u00b7 x<\/em>)<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\u00a0<\/strong>\r\n\r\nTerm<\/strong><\/td>\r\nA term can be a constant, a variable, or the product of a number and variable. (Terms are separated by a plus or minus sign.)<\/td>\r\n\r\n

\u00a0\u00a0\u00a0\u00a0 2x<\/em>3<\/sup> + 7x<\/em>2<\/sup> \u2013<\/em> 9y <\/em>\u2013<\/em> 8<\/p>\r\n

Terms: 2x<\/em>3<\/sup>, \u00a0\u00a07x<\/em>2<\/sup>,\u00a0 \u00a0- 9y,\u00a0 \u00a0-<\/em>8<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Like terms<\/strong><\/td>\r\nThe terms that have the same variables and exponents (differ only in their coefficients).<\/td>\r\n\r\n

2x<\/em> \u00a0\u00a0\u00a0\u00a0\u00a0and \u00a0\u00a0\u00a0\u00a0-7x<\/em><\/p>\r\n

-4y<\/em>2<\/sup>\u00a0\u00a0\u00a0 \u00a0\u00a0and \u00a0\u00a0\u00a0\u00a0\u00a09y<\/em>2<\/sup><\/p>\r\n

0.5pq<\/em>2<\/sup> \u00a0\u00a0 \u00a0\u00a0and\u00a0 \u00a0 \u00a0 $\\frac{2}{3}$pq<\/em>2<\/sup><\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\u00a0<\/strong>\r\n\r\nPolynomial<\/strong>: an algebraic expression that contains one or more terms.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/strong>\r\n

\r\n\r\nExample:<\/strong>\u00a0\u00a0\u00a0 7x ,\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0<\/em>5ax \u2013 <\/em>9b<\/em>\u00a0, \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/em>6x<\/em>2 \u2013<\/em> 5x <\/em>+ $\\frac{2}{3}$ ,\u00a0 \u00a0 \u00a0 \u00a07a<\/em>2<\/sup> + <\/em>8b + ab \u2013<\/em> 5\r\n\r\n<\/div>\r\n \r\n\r\nThere are special names for polynomials that have one, two, or three terms:<\/strong>\r\n
    \r\n \t
  • Mono<\/strong>mial: an algebraic expression that contains only one term.<\/li>\r\n<\/ul>\r\n
    \r\n\r\nExample<\/strong>:\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a09x ,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>4xy<\/em>2<\/sup> ,\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a00.8mn<\/em>2<\/sup> \u00a0,\u00a0 \u00a0 \u00a0 \u00a0$\\frac{1}{3}$a<\/em>2<\/sup>b<\/em>\r\n\r\n<\/div>\r\n
      \r\n \t
    • Bino<\/strong>mial: an algebraic expression that contains two terms.<\/li>\r\n<\/ul>\r\n
      \r\n

      Example<\/strong>:<\/strong> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 7x<\/em> + 9 ,\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a09t<\/em>2<\/sup> \u2013 2t ,\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>\u00a0\u00a0\u00a00.3y + <\/em>$\\frac{1}{3}$<\/p>\r\n\r\n<\/div>\r\n

        \r\n \t
      • Tri<\/strong>nomial: an algebraic expression that contains three terms.<\/li>\r\n<\/ul>\r\n
        \r\n\r\nExample:<\/strong> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ax<\/em>2<\/sup> + bx + c <\/em>,\u00a0 \u00a0 \u00a0 \u00a0\u2013 <\/em>4qp<\/em>2<\/sup> + 3q <\/em>+ 5\r\n\r\n<\/div>\r\n \r\n

        Combining Terms<\/strong><\/h2>\r\nLike terms:<\/strong> terms that have the same variables and exponents (the coefficients can be different).\r\n\r\nExamples:<\/strong>\r\n\r\n\r\n\r\n
        Example<\/strong><\/td>\r\nLike or unlike terms<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        7y<\/em>\u00a0 \u00a0and\u00a0 \u00a0-9y<\/em><\/td>\r\nLike terms<\/td>\r\n<\/tr>\r\n
        6a<\/em>2<\/sup>,\u00a0 \u00a0-32a<\/em>2<\/sup>,\u00a0 \u00a0and\u00a0\u00a0 -a<\/em>2<\/sup><\/td>\r\nLike terms<\/td>\r\n<\/tr>\r\n
        0.3 x<\/em>2<\/sup>y\u00a0 \u00a0<\/em>and\u00a0 \u00a0-48x2<\/sup>y<\/em><\/td>\r\nLike terms<\/td>\r\n<\/tr>\r\n
        $\\frac{-2}{7}$u<\/em>2<\/sup>v<\/em>3<\/sup>\u00a0 \u00a0and\u00a0 \u00a0u<\/em>2<\/sup>v<\/em>3<\/sup><\/td>\r\nLike terms<\/td>\r\n<\/tr>\r\n
        -8y<\/em>\u00a0 \u00a0and\u00a0 \u00a078x<\/em><\/td>\r\nUnlike terms<\/td>\r\n<\/tr>\r\n
        6m<\/em>3<\/sup>\u00a0 \u00a0and\u00a0 \u00a0-9m<\/em>2<\/sup><\/td>\r\nUnlike terms<\/td>\r\n<\/tr>\r\n
        -9u<\/em>3<\/sup>w<\/em>2<\/sup>\u00a0 \u00a0and\u00a0 \u00a0-9w<\/em>3<\/sup>u<\/em>2<\/sup><\/td>\r\nUnlike terms<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nCombine like terms:<\/strong> add or subtract their coefficients and keep the same variables and exponents.\r\n\r\nNote<\/strong>: unlike terms cannot be combined.\r\n
        \r\n

        Example 1.1.1<\/p>\r\n\r\n<\/header>\r\n

        \r\n\r\n\r\n\r\n\r\n
        1) <\/strong>3a<\/em> + 7b <\/em>\u2013 9a <\/em>+ 15b<\/em> = (3a<\/em> \u2013 9a<\/em>) + (7b<\/em> + 15b<\/em>)<\/td>\r\nRegroup like terms.<\/td>\r\n<\/tr>\r\n
        = -6a<\/em> + 22b<\/em><\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
        2<\/strong>) 2y2<\/sup> \u2013 4x + 3x \u2013 5y2<\/sup> = (2y2<\/sup> \u2013 5y2<\/sup>) + (-4x + 3x)<\/td>\r\nRegroup like terms.<\/td>\r\n<\/tr>\r\n
        = -3y2<\/sup> \u2013 1x<\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n
        = -3y2<\/sup> \u2013 x<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
        3) <\/strong>8xy2<\/sup><\/em>\u00a0\u2013 x<\/em>2<\/sup>y<\/em> + 4x2<\/sup><\/em>y <\/em>\u2013 6xy<\/em>2<\/sup><\/td>\r\n<\/td>\r\n<\/tr>\r\n
        = 8xy2<\/sup><\/em><\/span>\u00a0\u2013 x<\/em>2<\/sup>y<\/em><\/strong> + 4x<\/em>2<\/sup>y<\/em><\/strong> \u2013 6xy<\/em>2<\/sup><\/span><\/td>\r\nOr underline like terms without regrouping.<\/td>\r\n<\/tr>\r\n
        = 2xy2<\/sup><\/em>\u00a0+ 3x<\/em>2<\/sup>y<\/em><\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
        4) <\/strong>2(2m<\/em> + 3n<\/em>) + 3(m<\/em> \u2013 4n<\/em>) = 4m<\/em><\/span> + 6n<\/em><\/strong> + 3m<\/em><\/span> \u2013 12n<\/em><\/strong><\/td>\r\nDistributive property.<\/td>\r\n<\/tr>\r\n
        = 7m<\/em> \u2013 6n<\/em><\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
        \r\n\r\n \r\n

        Removing Parentheses<\/strong><\/h2>\r\n<\/div>\r\nIf the sign preceding the parentheses is positive (+),<\/strong> do not change the sign of terms inside the parentheses, just remove the parentheses.\r\n
        Example<\/strong>: \u00a0\u00a0\u00a0\u00a0(x<\/em> \u2013<\/em> 5) = x<\/em> \u2013<\/em> 5<\/div>\r\nIf the sign preceding the parentheses is negative (-),<\/strong> remove the parentheses and the negative sign (in front of parentheses), and change the sign of each term inside the parentheses.\r\n
        Example<\/strong>: \u00a0\u00a0\u00a0- (x<\/em> \u2013<\/em> 7) = -<\/em>x<\/em> + 7<\/div>\r\n \r\n\r\nRemove parentheses:\u00a0\u00a0<\/strong>\r\n\r\n\r\n\r\n
        Algebraic expression<\/strong><\/td>\r\nRemove parentheses<\/strong><\/td>\r\nExample<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        \u00a0 (ax<\/em> + b<\/em>)<\/td>\r\nax<\/em> + b<\/em><\/td>\r\n(5x<\/em> + 2) = 5x<\/em> + 2<\/td>\r\n<\/tr>\r\n
        \u00a0 (ax<\/em> \u2013<\/em> b<\/em>)<\/td>\r\nax<\/em> \u2013<\/em> b<\/em><\/td>\r\n(9y<\/em> \u2013 4) = 9y<\/em> \u2013 4<\/td>\r\n<\/tr>\r\n
        - (ax<\/em> + b<\/em>)<\/td>\r\n-ax<\/em> \u2013<\/em> b<\/em><\/td>\r\n- ($\\frac{3}{4}$x<\/em> + 7) = -<\/em> $\\frac{3}{4}$x<\/em> \u2013<\/em> 7<\/td>\r\n<\/tr>\r\n
        - (ax<\/em> \u2013<\/em> b<\/em>)<\/td>\r\n-ax<\/em> + b<\/em><\/td>\r\n- (0.5b<\/em> \u2013 2.4) = -0.5b<\/em> + 2.4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n
        \r\n

        Example 1.1.2<\/p>\r\n\r\n<\/header>\r\n

        \r\n\r\n\r\n\r\n\r\n
        1) <\/strong>\u00a09x2<\/sup><\/em>\u00a0+ 7 \u2013 (2x2<\/sup><\/em>\u00a0\u2013 2) = 9x2<\/sup><\/em><\/span>\u00a0+ 7<\/strong> \u2013 2x2<\/sup><\/em><\/span>\u00a0+ 2<\/strong><\/td>\r\nRemove parentheses.<\/td>\r\n<\/tr>\r\n
        = 7x2<\/sup><\/em>\u00a0+ 9<\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
        2)<\/strong> \u00a0(-8y<\/em> + 5z<\/em>) \u2013 4(y<\/em> \u2013 7z<\/em>) = -8y<\/em><\/span> + 5z<\/em><\/strong> \u2013 4y<\/em><\/span> + 28z<\/em><\/strong><\/td>\r\nRemove parentheses.<\/td>\r\n<\/tr>\r\n
        = -12y<\/em> + 33z<\/em><\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
        3) <\/strong>\u00a0\u00a0- (3a2<\/sup><\/em>\u00a0+ 4a <\/em>\u2013 4) + 3(4a2<\/sup><\/em>\u00a0\u2013 6a<\/em> + 7)<\/td>\r\nRemove parentheses.<\/td>\r\n<\/tr>\r\n
        = - 3a2<\/sup><\/em><\/strong>\u00a0\u2013 4a<\/em><\/span> + 4 + 12a2<\/sup><\/em><\/strong>\u00a0\u2013 18a<\/em><\/span> + 21<\/td>\r\nDistributive property.<\/td>\r\n<\/tr>\r\n
        = 9a2<\/sup><\/em>\u00a0\u2013 22a<\/em> + 25<\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
        4)<\/strong> \u00a0\u00a0-5(u2<\/sup><\/em>\u00a0\u2013<\/em> 3u<\/em>) + 3(2u<\/em> \u2013<\/em> 4) \u2013<\/em> (5 \u2013<\/em> 3u<\/em> +<\/em> 4u2<\/sup><\/em>)<\/td>\r\nDistributive property.<\/td>\r\n<\/tr>\r\n
        = -5u2<\/sup><\/em><\/strong>\u00a0+ 15u<\/em><\/span> + 6u<\/em><\/span> \u2013 12 \u2013 5 + 3u <\/em><\/span>\u2013 4u2<\/sup><\/em><\/strong><\/td>\r\nRemove parentheses.<\/td>\r\n<\/tr>\r\n
        = -9u2<\/sup><\/em>\u00a0+ 24u<\/em> \u2013 17<\/td>\r\nCombine like terms.<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
        \r\n\r\n \r\n

        Multiplying and Dividing Algebraic Expressions<\/strong><\/h2>\r\n<\/div>\r\nMultiplying a monomial and a polynomial:<\/strong>\r\n
          \r\n \t
        • Use the distributive property: a <\/em>(b<\/em> + c<\/em>) = ab<\/em> + ac<\/em><\/li>\r\n \t
        • Multiply coefficients and add exponents with the same base. Apply am<\/sup><\/em> an<\/sup><\/em> = am+n<\/sup><\/em><\/li>\r\n<\/ul>\r\n
          \r\n

          Example 1.1.3<\/p>\r\n\r\n<\/header>\r\n

          \r\n\r\n\r\n\r\n\r\n\r\n\r\n
          1)<\/strong> 3x<\/em>3<\/sup> (5x<\/em>2<\/sup> \u2013 2x<\/em>)<\/strong> = (3x<\/em>3<\/sup>) (5x<\/em>2<\/sup>) \u2013<\/strong> (3x<\/em>3<\/sup>) (2x<\/em>)<\/td>\r\nDistributive property:\u00a0 a <\/em>(b<\/em> + c<\/em>) = ab<\/em> + ac<\/em><\/td>\r\n<\/tr>\r\n
          = (3 \u2219 5) (x<\/em>3<\/sup> x<\/em>2<\/sup>) \u2013<\/strong> (3 \u2219 2) (x<\/em>3<\/sup> x<\/em>1<\/sup>)<\/td>\r\nRegroup\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x<\/em> = x<\/em>1<\/sup><\/td>\r\n<\/tr>\r\n
          = 15 (x<\/em>3+2<\/sup>) \u2013<\/strong> 6 (x<\/em>3+1<\/sup>)<\/td>\r\nMultiply the coefficients & add the exponents.<\/td>\r\n<\/tr>\r\n
          = 15x<\/em>5<\/sup> \u2013 <\/em>6x<\/em>4<\/sup><\/td>\r\nam<\/sup><\/em> \u2219 an<\/sup><\/em> = am+n<\/sup><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          2)<\/strong> 5ab<\/em>2<\/sup> (2a<\/em>2<\/sup>b +<\/em>ab<\/em>2<\/sup> \u2013 a<\/em>)<\/strong><\/td>\r\nDistribute.<\/td>\r\n<\/tr>\r\n
          = (5ab<\/em>2<\/sup>) (2a<\/em>2<\/sup>b<\/em>) +<\/em> (5ab<\/em>2<\/sup>) (ab<\/em>2<\/sup>) + (5ab<\/em>2<\/sup>) (-a<\/em>)<\/td>\r\nMultiply the coefficients and add exponents.<\/td>\r\n<\/tr>\r\n
          = (5 \u2219 2) (a<\/em>1+2<\/sup> b<\/em>2+1<\/sup>) +<\/em> (5a<\/em>1+1<\/sup> b<\/em>2+2<\/sup>) \u2013<\/strong> (5a<\/em>1+1<\/sup>b<\/em>2<\/sup>)<\/td>\r\nb<\/em> = b<\/em>1<\/sup>\u00a0\u00a0\u00a0\u00a0 ,\u00a0\u00a0\u00a0\u00a0 a<\/em> = a<\/em>1<\/sup><\/td>\r\n<\/tr>\r\n
          = 10a<\/em>3<\/sup>b<\/em>3<\/sup> + 5a<\/em>2<\/sup>b<\/em>4<\/sup> \u2013 5a<\/em>2<\/sup>b<\/em>2<\/sup><\/td>\r\nam<\/sup><\/em> \u2219 an<\/sup><\/em> = am+n<\/sup><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n \r\n\r\nDividing a polynomial by a monomial:<\/strong>\u00a0 <\/strong>\r\n
            \r\n \t
          • Split the polynomial into several parts.<\/li>\r\n \t
          • Divide a monomial by a monomial. Apply $\\frac{a^m}{a^n}=a^{m-n}$.<\/li>\r\n<\/ul>\r\n
            \r\n

            Example 1.1.4<\/p>\r\n\r\n<\/header>\r\n

            \r\n

            $\\frac{12x^2+4x-2}{4x}$<\/p>\r\n

            Steps\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Solution<\/strong><\/p>\r\n\r\n

              \r\n \t
            • Split the polynomial into three parts:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 $\\frac{12x^2+4x-2}{4x}=\\frac{12x^2}{4x}+\\frac{4x}{4x}-\\frac{2}{4x}$<\/li>\r\n \t
            • Divide a monomial by a monomial:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 $=3x+1-\\frac{1}{2x}$ \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/strong>$\\frac{a^m}{a^n}=a^{m-n}$<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n
              \r\n\r\n\u00a0<\/strong>\r\n\r\nThe FOIL method:<\/strong>\u00a0an easy way to find the product of two binomials (two terms).<\/span>\r\n\r\n<\/div>\r\n\r\n\r\n\r\n
              (a<\/em> + b<\/em>) (c<\/em> + d<\/em>) = ac<\/em> + ad<\/em> + bc<\/em> + bd<\/em><\/strong>\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0F\u00a0 \u00a0 \u00a0 O\u00a0\u00a0\u00a0\u00a0\u00a0 I<\/strong>\u00a0 \u00a0 \u00a0 L<\/strong><\/td>\r\nExample<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
              F<\/strong> - First terms<\/td>\r\nfirst term \u00d7 first term\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(a<\/em><\/strong><\/span> + b<\/em>) (c<\/em><\/strong><\/span> + d<\/em>)<\/td>\r\n(x<\/em><\/strong><\/span> + 5) (x<\/em> <\/strong><\/span>+ 4)<\/td>\r\n<\/tr>\r\n
              O<\/strong> - Outer terms<\/td>\r\noutside term \u00d7 outside term\u00a0 \u00a0 \u00a0(a<\/em><\/strong><\/span> + b<\/em>) (c<\/em> + d<\/em><\/strong><\/span>)<\/td>\r\n(x<\/em><\/strong><\/span> + 5) (x<\/em> + 4<\/strong><\/span>)<\/td>\r\n<\/tr>\r\n
              I<\/strong> - Inner terms<\/td>\r\ninside term \u00d7 inside term\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(a<\/em> + b<\/em><\/strong><\/span>) (c<\/em><\/strong><\/span> + d<\/em>)<\/td>\r\n(x<\/em> + 5<\/strong><\/span>) (x<\/em><\/strong><\/span> + 4)<\/td>\r\n<\/tr>\r\n
              L<\/strong> - Last terms<\/td>\r\nlast term \u00d7 last term\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(a<\/em> + b<\/em><\/strong><\/span>) (c<\/em> + d<\/em><\/strong><\/span>)<\/td>\r\n(x<\/em> + 5<\/strong><\/span>) (x<\/em> + 4<\/strong><\/span>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n\r\n\r\n
              FOIL method<\/strong><\/td>\r\nExample<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
              (a<\/em> + b<\/em>) (c<\/em> + d<\/em>) = ac<\/em> + ad<\/em> + bc<\/em> + bd<\/em><\/td>\r\n(x<\/em> + 5) (x<\/em> + 4) = x \u2219 x<\/em> + x \u2219 <\/em>4 + 5x<\/em> + 5 \u2219 4 = x<\/em>2<\/sup> + 9x<\/em> + 20<\/td>\r\n<\/tr>\r\n
              \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0F\u00a0\u00a0\u00a0\u00a0 O\u00a0\u00a0\u00a0\u00a0\u00a0 I\u00a0 \u00a0 \u00a0 \u00a0L<\/strong><\/td>\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0F\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 O\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 I<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 L<\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\u00a0<\/strong>\r\n\r\nMultiplying binomials (2 terms \u00d7 2 terms):<\/strong>\u00a0 <\/strong>\r\n
              \r\n

              Example 1.1.5<\/p>\r\n\r\n<\/header>\r\n

              \r\n\r\n\r\n\r\n\r\n\r\n\r\n
              1) <\/strong>$(2x + 3)(5x - 6)=2x \\cdot 5x+2x(- 6)+3 \\cdot 5x+3(-6)$\r\n\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 F\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0O\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0I\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0L\u00a0<\/strong><\/td>\r\nThe FOIL method.<\/td>\r\n<\/tr>\r\n
              <\/td>\r\n<\/tr>\r\n
              $=10x^2-12x+15x-18$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
              $=10x^2+3x-18$<\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n\r\n\r\n\r\n\r\n
              2) <\/strong>$(3r \u2013 t)(5r+t^2)=3r \\cdot 5r+3r \\cdot t^2-t \\cdot 5r-t \\cdot t^2$<\/td>\r\nFOIL<\/td>\r\n<\/tr>\r\n
              $=15r^2+3rt^2-5rt-t^3$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
              $=18r^2-5rt-t^3$<\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n\r\n\r\n\r\n\r\n
              3) <\/strong>$(xy^2+y)(2x^2y+x)=xy^2 \\cdot 2x^2y+xy^2 \\cdot x+y \\cdot 2x^2y+yx$<\/td>\r\nFOIL<\/td>\r\n<\/tr>\r\n
              $=2x^3y^3+x^2y^2+2x^2y^2+xy$<\/td>\r\n<\/td>\r\n<\/tr>\r\n
              $=2x^3y^3+3x^2y^2+xy$<\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n\r\n\r\n\r\n
              4) <\/strong>$(a-\\frac{1}{3})(a-\\frac{1}{3})=a^2-\\frac{1}{3}a-\\frac{1}{3}a+(-\\frac{1}{3})(-\\frac{1}{3})$<\/td>\r\nFOIL<\/td>\r\n<\/tr>\r\n
              $=a^2-\\frac{2}{3}a+\\frac{1}{9}$<\/td>\r\nCombine like terms.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n
              \r\n

              Practice questions<\/h2>\r\n<\/div>\r\n1.<\/strong> Identify the terms of each polynomial:\r\n

              a. <\/strong>5x<\/em>3<\/sup> + 8x<\/em>2<\/sup> + 2x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em><\/p>\r\n

              b. <\/strong>\u2013$\\frac{2}{3}$y<\/em>4<\/sup> + 9a<\/em>2<\/sup> + a <\/em>\u2013 1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em><\/p>\r\n2.<\/strong> Combine like terms:\r\n

              a. <\/strong>7x <\/em>+ 10y <\/em>\u2013 8x <\/em>+ 9y<\/em><\/p>\r\n

              b. <\/strong>12a2<\/sup><\/em>\u00a0\u2013<\/em> 33b <\/em>+ 2b<\/em> \u2013 6a2<\/sup><\/em><\/p>\r\n

              c. <\/strong>13n<\/em> + 5(6n<\/em> \u2013 m<\/em>2<\/sup>) +<\/strong> 7(2m<\/em>2<\/sup> + 3n<\/em>)<\/p>\r\n3.<\/strong> Simplify:\r\n

              a. <\/strong>15a2<\/sup><\/em>\u00a0+ 9 \u2013 (5a2<\/sup><\/em>\u00a0\u2013 4)<\/p>\r\n

              b. <\/strong>(-13x<\/em> + 9y<\/em>) \u2013 6(x<\/em> \u2013 5y<\/em>)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/strong><\/p>\r\n

              c.<\/strong>\u00a0<\/strong>5(<\/span>ab<\/em> \u2013 2<\/span>xy<\/em>) \u2013 6(-2<\/span>ab<\/em> + 3<\/span>xy<\/em>)<\/span><\/p>\r\n

              d. <\/strong>(5y<\/em> \u2013 7) (8y<\/em> + 9)<\/p>\r\n

              e. <\/strong>(7r<\/em> \u2013 2t<\/em>) (3r<\/em> + 4t<\/em>2<\/sup>)<\/p>\r\n

              f.<\/strong>\u00a0(x <\/em>\u2013 $\\frac{1}{3}$) (x<\/em> \u2013 $\\frac{2}{3}$)<\/p>\r\n ","rendered":"

              \n

              Introduction to Algebra<\/h1>\n<\/div>\n

              Review of basic algebraic terms:<\/strong><\/p>\n\n\n\n
              Algebraic term<\/strong><\/td>\nDescription<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n\n\n\n
              \u00a0Algebraic expression<\/strong><\/td>\nA mathematical phrase that contains numbers, variables (letters), and arithmetic operations (+, \u2013<\/em> , \u00d7, \u00f7, etc.).<\/td>\n\n

              3x<\/em> \u2013 4<\/p>\n

              5a<\/em>2<\/sup> \u2013 <\/em>b + <\/em>3<\/p>\n

              12y<\/em>3<\/sup> + 7y<\/em>2<\/sup> \u2013<\/em> 5y <\/em>+ \"\frac{2}{3}\"<\/p>\n<\/td>\n<\/tr>\n

              Constant<\/strong><\/td>\nA number on its own.<\/td>\n2y + 5\u00a0 \u00a0 \u00a0 \u00a0 constant:\u00a0 \u00a0 \u00a0 \u00a0 5<\/td>\n<\/tr>\n
              Coefficient<\/strong><\/td>\nThe number in front of a variable.<\/td>\n\n

              -9x2<\/sup>\u00a0 \u00a0 <\/em>coefficient:\u00a0 \u00a0 -9<\/p>\n

              \u00a0x<\/em>\u00a0 \u00a0 coefficient:\u00a0 \u00a0\u00a01<\/p>\n

              (x<\/em> = 1 \u00b7 x<\/em>)<\/p>\n<\/td>\n<\/tr>\n

              \u00a0<\/strong><\/p>\n

              Term<\/strong><\/td>\n

              		A term can be a constant, a variable, or the product of a number and variable. (Terms are separated by a plus or minus sign.)<\/td>\n\n

              \u00a0\u00a0\u00a0\u00a0 2x<\/em>3<\/sup> + 7x<\/em>2<\/sup> \u2013<\/em> 9y <\/em>\u2013<\/em> 8<\/p>\n

              Terms: 2x<\/em>3<\/sup>, \u00a0\u00a07x<\/em>2<\/sup>,\u00a0 \u00a0– 9y,\u00a0 \u00a0–<\/em>8<\/p>\n<\/td>\n<\/tr>\n

              Like terms<\/strong><\/td>\nThe terms that have the same variables and exponents (differ only in their coefficients).<\/td>\n\n

              2x<\/em> \u00a0\u00a0\u00a0\u00a0\u00a0and \u00a0\u00a0\u00a0\u00a0-7x<\/em><\/p>\n

              -4y<\/em>2<\/sup>\u00a0\u00a0\u00a0 \u00a0\u00a0and \u00a0\u00a0\u00a0\u00a0\u00a09y<\/em>2<\/sup><\/p>\n

              0.5pq<\/em>2<\/sup> \u00a0\u00a0 \u00a0\u00a0and\u00a0 \u00a0 \u00a0 \"\frac{2}{3}\"pq<\/em>2<\/sup><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

              \u00a0<\/strong><\/p>\n

              Polynomial<\/strong>: an algebraic expression that contains one or more terms.\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/strong><\/p>\n

              \n

              Example:<\/strong>\u00a0\u00a0\u00a0 7x ,\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0<\/em>5ax \u2013 <\/em>9b<\/em>\u00a0, \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/em>6x<\/em>2 \u2013<\/em> 5x <\/em>+ \"\frac{2}{3}\" ,\u00a0 \u00a0 \u00a0 \u00a07a<\/em>2<\/sup> + <\/em>8b + ab \u2013<\/em> 5<\/p>\n<\/div>\n

               <\/p>\n

              There are special names for polynomials that have one, two, or three terms:<\/strong><\/p>\n

                \n
              • Mono<\/strong>mial: an algebraic expression that contains only one term.<\/li>\n<\/ul>\n
                \n

                Example<\/strong>:\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a09x ,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>4xy<\/em>2<\/sup> ,\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a00.8mn<\/em>2<\/sup> \u00a0,\u00a0 \u00a0 \u00a0 \u00a0\"\frac{1}{3}\"a<\/em>2<\/sup>b<\/em><\/p>\n<\/div>\n

                  \n
                • Bino<\/strong>mial: an algebraic expression that contains two terms.<\/li>\n<\/ul>\n
                  \n

                  Example<\/strong>:<\/strong> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 7x<\/em> + 9 ,\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a09t<\/em>2<\/sup> \u2013 2t ,\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em>\u00a0\u00a0\u00a00.3y + <\/em>\"\frac{1}{3}\"<\/p>\n<\/div>\n

                    \n
                  • Tri<\/strong>nomial: an algebraic expression that contains three terms.<\/li>\n<\/ul>\n
                    \n

                    Example:<\/strong> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ax<\/em>2<\/sup> + bx + c <\/em>,\u00a0 \u00a0 \u00a0 \u00a0\u2013 <\/em>4qp<\/em>2<\/sup> + 3q <\/em>+ 5<\/p>\n<\/div>\n

                     <\/p>\n

                    Combining Terms<\/strong><\/h2>\n

                    Like terms:<\/strong> terms that have the same variables and exponents (the coefficients can be different).<\/p>\n

                    Examples:<\/strong><\/p>\n\n\n\n
                    Example<\/strong><\/td>\nLike or unlike terms<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n\n\n\n\n\n
                    7y<\/em>\u00a0 \u00a0and\u00a0 \u00a0-9y<\/em><\/td>\nLike terms<\/td>\n<\/tr>\n
                    6a<\/em>2<\/sup>,\u00a0 \u00a0-32a<\/em>2<\/sup>,\u00a0 \u00a0and\u00a0\u00a0 –a<\/em>2<\/sup><\/td>\nLike terms<\/td>\n<\/tr>\n
                    0.3 x<\/em>2<\/sup>y\u00a0 \u00a0<\/em>and\u00a0 \u00a0-48x2<\/sup>y<\/em><\/td>\nLike terms<\/td>\n<\/tr>\n
                    \"\frac{-2}{7}\"u<\/em>2<\/sup>v<\/em>3<\/sup>\u00a0 \u00a0and\u00a0 \u00a0u<\/em>2<\/sup>v<\/em>3<\/sup><\/td>\nLike terms<\/td>\n<\/tr>\n
                    -8y<\/em>\u00a0 \u00a0and\u00a0 \u00a078x<\/em><\/td>\nUnlike terms<\/td>\n<\/tr>\n
                    6m<\/em>3<\/sup>\u00a0 \u00a0and\u00a0 \u00a0-9m<\/em>2<\/sup><\/td>\nUnlike terms<\/td>\n<\/tr>\n
                    -9u<\/em>3<\/sup>w<\/em>2<\/sup>\u00a0 \u00a0and\u00a0 \u00a0-9w<\/em>3<\/sup>u<\/em>2<\/sup><\/td>\nUnlike terms<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                     <\/p>\n

                    Combine like terms:<\/strong> add or subtract their coefficients and keep the same variables and exponents.<\/p>\n

                    Note<\/strong>: unlike terms cannot be combined.<\/p>\n

                    \n
                    \n

                    Example 1.1.1<\/p>\n<\/header>\n

                    \n\n\n\n\n
                    1) <\/strong>3a<\/em> + 7b <\/em>\u2013 9a <\/em>+ 15b<\/em> = (3a<\/em> \u2013 9a<\/em>) + (7b<\/em> + 15b<\/em>)<\/td>\nRegroup like terms.<\/td>\n<\/tr>\n
                    = -6a<\/em> + 22b<\/em><\/td>\nCombine like terms.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n
                    2<\/strong>) 2y2<\/sup> \u2013 4x + 3x \u2013 5y2<\/sup> = (2y2<\/sup> \u2013 5y2<\/sup>) + (-4x + 3x)<\/td>\nRegroup like terms.<\/td>\n<\/tr>\n
                    = -3y2<\/sup> \u2013 1x<\/td>\nCombine like terms.<\/td>\n<\/tr>\n
                    = -3y2<\/sup> \u2013 x<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n
                    3) <\/strong>8xy2<\/sup><\/em>\u00a0\u2013 x<\/em>2<\/sup>y<\/em> + 4x2<\/sup><\/em>y <\/em>\u2013 6xy<\/em>2<\/sup><\/td>\n<\/td>\n<\/tr>\n
                    = 8xy2<\/sup><\/em><\/span>\u00a0\u2013 x<\/em>2<\/sup>y<\/em><\/strong> + 4x<\/em>2<\/sup>y<\/em><\/strong> \u2013 6xy<\/em>2<\/sup><\/span><\/td>\nOr underline like terms without regrouping.<\/td>\n<\/tr>\n
                    = 2xy2<\/sup><\/em>\u00a0+ 3x<\/em>2<\/sup>y<\/em><\/td>\nCombine like terms.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
                    4) <\/strong>2(2m<\/em> + 3n<\/em>) + 3(m<\/em> \u2013 4n<\/em>) = 4m<\/em><\/span> + 6n<\/em><\/strong> + 3m<\/em><\/span> \u2013 12n<\/em><\/strong><\/td>\nDistributive property.<\/td>\n<\/tr>\n
                    = 7m<\/em> \u2013 6n<\/em><\/td>\nCombine like terms.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
                    \n

                     <\/p>\n

                    Removing Parentheses<\/strong><\/h2>\n<\/div>\n

                    If the sign preceding the parentheses is positive (+),<\/strong> do not change the sign of terms inside the parentheses, just remove the parentheses.<\/p>\n

                    Example<\/strong>: \u00a0\u00a0\u00a0\u00a0(x<\/em> \u2013<\/em> 5) = x<\/em> \u2013<\/em> 5<\/div>\n

                    If the sign preceding the parentheses is negative (-),<\/strong> remove the parentheses and the negative sign (in front of parentheses), and change the sign of each term inside the parentheses.<\/p>\n

                    Example<\/strong>: \u00a0\u00a0\u00a0– (x<\/em> \u2013<\/em> 7) = –<\/em>x<\/em> + 7<\/div>\n

                     <\/p>\n

                    Remove parentheses:\u00a0\u00a0<\/strong><\/p>\n\n\n\n
                    Algebraic expression<\/strong><\/td>\nRemove parentheses<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n\n\n
                    \u00a0 (ax<\/em> + b<\/em>)<\/td>\nax<\/em> + b<\/em><\/td>\n(5x<\/em> + 2) = 5x<\/em> + 2<\/td>\n<\/tr>\n
                    \u00a0 (ax<\/em> \u2013<\/em> b<\/em>)<\/td>\nax<\/em> \u2013<\/em> b<\/em><\/td>\n(9y<\/em> \u2013 4) = 9y<\/em> \u2013 4<\/td>\n<\/tr>\n
                    – (ax<\/em> + b<\/em>)<\/td>\n-ax<\/em> \u2013<\/em> b<\/em><\/td>\n– (\"\frac{3}{4}\"x<\/em> + 7) = –<\/em> \"\frac{3}{4}\"x<\/em> \u2013<\/em> 7<\/td>\n<\/tr>\n
                    – (ax<\/em> \u2013<\/em> b<\/em>)<\/td>\n-ax<\/em> + b<\/em><\/td>\n– (0.5b<\/em> \u2013 2.4) = -0.5b<\/em> + 2.4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                     <\/p>\n

                    \n
                    \n

                    Example 1.1.2<\/p>\n<\/header>\n

                    \n\n\n\n\n
                    1) <\/strong>\u00a09x2<\/sup><\/em>\u00a0+ 7 \u2013 (2x2<\/sup><\/em>\u00a0\u2013 2) = 9x2<\/sup><\/em><\/span>\u00a0+ 7<\/strong> \u2013 2x2<\/sup><\/em><\/span>\u00a0+ 2<\/strong><\/td>\nRemove parentheses.<\/td>\n<\/tr>\n
                    = 7x2<\/sup><\/em>\u00a0+ 9<\/td>\nCombine like terms.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n
                    2)<\/strong> \u00a0(-8y<\/em> + 5z<\/em>) \u2013 4(y<\/em> \u2013 7z<\/em>) = -8y<\/em><\/span> + 5z<\/em><\/strong> \u2013 4y<\/em><\/span> + 28z<\/em><\/strong><\/td>\nRemove parentheses.<\/td>\n<\/tr>\n
                    = -12y<\/em> + 33z<\/em><\/td>\nCombine like terms.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n
                    3) <\/strong>\u00a0\u00a0– (3a2<\/sup><\/em>\u00a0+ 4a <\/em>\u2013 4) + 3(4a2<\/sup><\/em>\u00a0\u2013 6a<\/em> + 7)<\/td>\nRemove parentheses.<\/td>\n<\/tr>\n
                    = – 3a2<\/sup><\/em><\/strong>\u00a0\u2013 4a<\/em><\/span> + 4 + 12a2<\/sup><\/em><\/strong>\u00a0\u2013 18a<\/em><\/span> + 21<\/td>\nDistributive property.<\/td>\n<\/tr>\n
                    = 9a2<\/sup><\/em>\u00a0\u2013 22a<\/em> + 25<\/td>\nCombine like terms.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n
                    4)<\/strong> \u00a0\u00a0-5(u2<\/sup><\/em>\u00a0\u2013<\/em> 3u<\/em>) + 3(2u<\/em> \u2013<\/em> 4) \u2013<\/em> (5 \u2013<\/em> 3u<\/em> +<\/em> 4u2<\/sup><\/em>)<\/td>\nDistributive property.<\/td>\n<\/tr>\n
                    = -5u2<\/sup><\/em><\/strong>\u00a0+ 15u<\/em><\/span> + 6u<\/em><\/span> \u2013 12 \u2013 5 + 3u <\/em><\/span>\u2013 4u2<\/sup><\/em><\/strong><\/td>\nRemove parentheses.<\/td>\n<\/tr>\n
                    = -9u2<\/sup><\/em>\u00a0+ 24u<\/em> \u2013 17<\/td>\nCombine like terms.<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
                    \n

                     <\/p>\n

                    Multiplying and Dividing Algebraic Expressions<\/strong><\/h2>\n<\/div>\n

                    Multiplying a monomial and a polynomial:<\/strong><\/p>\n

                      \n
                    • Use the distributive property: a <\/em>(b<\/em> + c<\/em>) = ab<\/em> + ac<\/em><\/li>\n
                    • Multiply coefficients and add exponents with the same base. Apply am<\/sup><\/em> an<\/sup><\/em> = am+n<\/sup><\/em><\/li>\n<\/ul>\n
                      \n
                      \n

                      Example 1.1.3<\/p>\n<\/header>\n

                      \n\n\n\n\n\n\n
                      1)<\/strong> 3x<\/em>3<\/sup> (5x<\/em>2<\/sup> \u2013 2x<\/em>)<\/strong> = (3x<\/em>3<\/sup>) (5x<\/em>2<\/sup>) \u2013<\/strong> (3x<\/em>3<\/sup>) (2x<\/em>)<\/td>\nDistributive property:\u00a0 a <\/em>(b<\/em> + c<\/em>) = ab<\/em> + ac<\/em><\/td>\n<\/tr>\n
                      = (3 \u2219 5) (x<\/em>3<\/sup> x<\/em>2<\/sup>) \u2013<\/strong> (3 \u2219 2) (x<\/em>3<\/sup> x<\/em>1<\/sup>)<\/td>\nRegroup\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x<\/em> = x<\/em>1<\/sup><\/td>\n<\/tr>\n
                      = 15 (x<\/em>3+2<\/sup>) \u2013<\/strong> 6 (x<\/em>3+1<\/sup>)<\/td>\nMultiply the coefficients & add the exponents.<\/td>\n<\/tr>\n
                      = 15x<\/em>5<\/sup> \u2013 <\/em>6x<\/em>4<\/sup><\/td>\nam<\/sup><\/em> \u2219 an<\/sup><\/em> = am+n<\/sup><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n
                      2)<\/strong> 5ab<\/em>2<\/sup> (2a<\/em>2<\/sup>b +<\/em>ab<\/em>2<\/sup> \u2013 a<\/em>)<\/strong><\/td>\nDistribute.<\/td>\n<\/tr>\n
                      = (5ab<\/em>2<\/sup>) (2a<\/em>2<\/sup>b<\/em>) +<\/em> (5ab<\/em>2<\/sup>) (ab<\/em>2<\/sup>) + (5ab<\/em>2<\/sup>) (-a<\/em>)<\/td>\nMultiply the coefficients and add exponents.<\/td>\n<\/tr>\n
                      = (5 \u2219 2) (a<\/em>1+2<\/sup> b<\/em>2+1<\/sup>) +<\/em> (5a<\/em>1+1<\/sup> b<\/em>2+2<\/sup>) \u2013<\/strong> (5a<\/em>1+1<\/sup>b<\/em>2<\/sup>)<\/td>\nb<\/em> = b<\/em>1<\/sup>\u00a0\u00a0\u00a0\u00a0 ,\u00a0\u00a0\u00a0\u00a0 a<\/em> = a<\/em>1<\/sup><\/td>\n<\/tr>\n
                      = 10a<\/em>3<\/sup>b<\/em>3<\/sup> + 5a<\/em>2<\/sup>b<\/em>4<\/sup> \u2013 5a<\/em>2<\/sup>b<\/em>2<\/sup><\/td>\nam<\/sup><\/em> \u2219 an<\/sup><\/em> = am+n<\/sup><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

                       <\/p>\n

                      Dividing a polynomial by a monomial:<\/strong>\u00a0 <\/strong><\/p>\n

                        \n
                      • Split the polynomial into several parts.<\/li>\n
                      • Divide a monomial by a monomial. Apply \"\frac{a^m}{a^n}=a^{m-n}\".<\/li>\n<\/ul>\n
                        \n
                        \n

                        Example 1.1.4<\/p>\n<\/header>\n

                        \n

                        \"\frac{12x^2+4x-2}{4x}\"<\/p>\n

                        Steps\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Solution<\/strong><\/p>\n

                          \n
                        • Split the polynomial into three parts:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \"\frac{12x^2+4x-2}{4x}=\frac{12x^2}{4x}+\frac{4x}{4x}-\frac{2}{4x}\"<\/li>\n
                        • Divide a monomial by a monomial:\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \"=3x+1-\frac{1}{2x}\" \u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/strong>\"\frac{a^m}{a^n}=a^{m-n}\"<\/li>\n<\/ul>\n<\/div>\n<\/div>\n
                          \n

                          \u00a0<\/strong><\/p>\n

                          The FOIL method:<\/strong>\u00a0an easy way to find the product of two binomials (two terms).<\/span><\/p>\n<\/div>\n\n\n\n
                          (a<\/em> + b<\/em>) (c<\/em> + d<\/em>) = ac<\/em> + ad<\/em> + bc<\/em> + bd<\/em><\/strong>
                          \n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0F\u00a0 \u00a0 \u00a0 O\u00a0\u00a0\u00a0\u00a0\u00a0 I<\/strong>\u00a0 \u00a0 \u00a0 L<\/strong><\/td>\n                          		Example<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n\n\n
                          F<\/strong> – First terms<\/td>\nfirst term \u00d7 first term\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(a<\/em><\/strong><\/span> + b<\/em>) (c<\/em><\/strong><\/span> + d<\/em>)<\/td>\n(x<\/em><\/strong><\/span> + 5) (x<\/em> <\/strong><\/span>+ 4)<\/td>\n<\/tr>\n
                          O<\/strong> – Outer terms<\/td>\noutside term \u00d7 outside term\u00a0 \u00a0 \u00a0(a<\/em><\/strong><\/span> + b<\/em>) (c<\/em> + d<\/em><\/strong><\/span>)<\/td>\n(x<\/em><\/strong><\/span> + 5) (x<\/em> + 4<\/strong><\/span>)<\/td>\n<\/tr>\n
                          I<\/strong> – Inner terms<\/td>\ninside term \u00d7 inside term\u00a0 \u00a0 \u00a0 \u00a0 \u00a0(a<\/em> + b<\/em><\/strong><\/span>) (c<\/em><\/strong><\/span> + d<\/em>)<\/td>\n(x<\/em> + 5<\/strong><\/span>) (x<\/em><\/strong><\/span> + 4)<\/td>\n<\/tr>\n
                          L<\/strong> – Last terms<\/td>\nlast term \u00d7 last term\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(a<\/em> + b<\/em><\/strong><\/span>) (c<\/em> + d<\/em><\/strong><\/span>)<\/td>\n(x<\/em> + 5<\/strong><\/span>) (x<\/em> + 4<\/strong><\/span>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                           <\/p>\n\n\n\n
                          FOIL method<\/strong><\/td>\nExample<\/strong><\/td>\n<\/tr>\n<\/thead>\n<\/table>\n\n\n\n\n
                          (a<\/em> + b<\/em>) (c<\/em> + d<\/em>) = ac<\/em> + ad<\/em> + bc<\/em> + bd<\/em><\/td>\n(x<\/em> + 5) (x<\/em> + 4) = x \u2219 x<\/em> + x \u2219 <\/em>4 + 5x<\/em> + 5 \u2219 4 = x<\/em>2<\/sup> + 9x<\/em> + 20<\/td>\n<\/tr>\n
                          \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0F\u00a0\u00a0\u00a0\u00a0 O\u00a0\u00a0\u00a0\u00a0\u00a0 I\u00a0 \u00a0 \u00a0 \u00a0L<\/strong><\/td>\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0F\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 O\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 I<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 L<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                          \u00a0<\/strong><\/p>\n

                          Multiplying binomials (2 terms \u00d7 2 terms):<\/strong>\u00a0 <\/strong><\/p>\n

                          \n
                          \n

                          Example 1.1.5<\/p>\n<\/header>\n

                          \n\n\n\n\n\n\n
                          1) <\/strong>\"(2x + 3)(5x - 6)=2x \cdot 5x+2x(- 6)+3 \cdot 5x+3(-6)\"<\/p>\n

                          \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 F\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0O\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0I\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0L\u00a0<\/strong><\/td>\n

                          		The FOIL method.<\/td>\n<\/tr>\n
                          <\/td>\n<\/tr>\n
                          \"=10x^2-12x+15x-18\"<\/td>\n<\/td>\n<\/tr>\n
                          \"=10x^2+3x-18\"<\/td>\nCombine like terms.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                           <\/p>\n\n\n\n\n\n
                          2) <\/strong>\"(3r - t)(5r+t^2)=3r \cdot 5r+3r \cdot t^2-t \cdot 5r-t \cdot t^2\"<\/td>\nFOIL<\/td>\n<\/tr>\n
                          \"=15r^2+3rt^2-5rt-t^3\"<\/td>\n<\/td>\n<\/tr>\n
                          \"=18r^2-5rt-t^3\"<\/td>\nCombine like terms.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                           <\/p>\n\n\n\n\n\n
                          3) <\/strong>\"(xy^2+y)(2x^2y+x)=xy^2 \cdot 2x^2y+xy^2 \cdot x+y \cdot 2x^2y+yx\"<\/td>\nFOIL<\/td>\n<\/tr>\n
                          \"=2x^3y^3+x^2y^2+2x^2y^2+xy\"<\/td>\n<\/td>\n<\/tr>\n
                          \"=2x^3y^3+3x^2y^2+xy\"<\/td>\nCombine like terms.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                           <\/p>\n\n\n\n\n
                          4) <\/strong>\"(a-\frac{1}{3})(a-\frac{1}{3})=a^2-\frac{1}{3}a-\frac{1}{3}a+(-\frac{1}{3})(-\frac{1}{3})\"<\/td>\nFOIL<\/td>\n<\/tr>\n
                          \"=a^2-\frac{2}{3}a+\frac{1}{9}\"<\/td>\nCombine like terms.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n
                          \n

                          Practice questions<\/h2>\n<\/div>\n

                          1.<\/strong> Identify the terms of each polynomial:<\/p>\n

                          a. <\/strong>5x<\/em>3<\/sup> + 8x<\/em>2<\/sup> + 2x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em><\/p>\n

                          b. <\/strong>\u2013\"\frac{2}{3}\"y<\/em>4<\/sup> + 9a<\/em>2<\/sup> + a <\/em>\u2013 1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/em><\/p>\n

                          2.<\/strong> Combine like terms:<\/p>\n

                          a. <\/strong>7x <\/em>+ 10y <\/em>\u2013 8x <\/em>+ 9y<\/em><\/p>\n

                          b. <\/strong>12a2<\/sup><\/em>\u00a0\u2013<\/em> 33b <\/em>+ 2b<\/em> \u2013 6a2<\/sup><\/em><\/p>\n

                          c. <\/strong>13n<\/em> + 5(6n<\/em> \u2013 m<\/em>2<\/sup>) +<\/strong> 7(2m<\/em>2<\/sup> + 3n<\/em>)<\/p>\n

                          3.<\/strong> Simplify:<\/p>\n

                          a. <\/strong>15a2<\/sup><\/em>\u00a0+ 9 \u2013 (5a2<\/sup><\/em>\u00a0\u2013 4)<\/p>\n

                          b. <\/strong>(-13x<\/em> + 9y<\/em>) \u2013 6(x<\/em> \u2013 5y<\/em>)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/strong><\/p>\n

                          c.<\/strong>\u00a0<\/strong>5(<\/span>ab<\/em> \u2013 2<\/span>xy<\/em>) \u2013 6(-2<\/span>ab<\/em> + 3<\/span>xy<\/em>)<\/span><\/p>\n

                          d. <\/strong>(5y<\/em> \u2013 7) (8y<\/em> + 9)<\/p>\n

                          e. <\/strong>(7r<\/em> \u2013 2t<\/em>) (3r<\/em> + 4t<\/em>2<\/sup>)<\/p>\n

                          f.<\/strong>\u00a0(x <\/em>\u2013 \"\frac{1}{3}\") (x<\/em> \u2013 \"\frac{2}{3}\")<\/p>\n

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