1.1. Introduction to Algebra

Ian Young

1.1. Introduction to Algebra

Introduction

Review of basic algebraic terms:

Algebraic term	Description	Example

Algebraic expression	A mathematical phrase that contains numbers, variables (letters), and arithmetic operations (+, – , ×, ÷, etc.).	3x – 4 5a² – b + 3 12y³ + 7y² – 5y + $\frac{2}{3}$
Constant	A number on its own.	2y + 5 constant: 5
Coefficient	The number in front of a variable.	-9x² coefficient: -9 x coefficient: 1 (x = 1 · x)
Term	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.)	2x³ + 7x² – 9y – 8 Terms: 2x³, 7x², – 9y, –8
Like terms	The terms that have the same variables and exponents (differ only in their coefficients).	2x and -7x -4y² and 9y² 0.5pq² and $\frac{2}{3}$ pq²

Polynomial: an algebraic expression that contains one or more terms.

Example: 7x , 5ax – 9b , 6x2 – 5x + $\frac{2}{3}$ , 7a² + 8b + ab – 5

There are special names for polynomials that have one, two, or three terms:

Monomial: an algebraic expression that contains only one term.

Example: 9x , 4xy² , 0.8mn² , $\frac{1}{3}$ a²b

Binomial: an algebraic expression that contains two terms.

Example: 7x + 9 , 9t² – 2t , 0.3y + $\frac{1}{3}$

Trinomial: an algebraic expression that contains three terms.

Example: ax² + bx + c , – 4qp² + 3q + 5

Combining Terms

Like terms: terms that have the same variables and exponents (the coefficients can be different).

Examples:

Example	Like or unlike terms

7y and -9y	Like terms
6a², -32a², and –a²	Like terms
0.3 x²y and -48x²y	Like terms
$\frac{-2}{7}$ u²v³ and u²v³	Like terms
-8y and 78x	Unlike terms
6m³ and -9m²	Unlike terms
-9u³w² and -9w³u²	Unlike terms

Combine like terms: add or subtract their coefficients and keep the same variables and exponents.

Note: unlike terms cannot be combined.

Example 1.1.1

1) 3a + 7b – 9a + 15b = (3a – 9a) + (7b + 15b)	Regroup like terms.
= -6a + 22b	Combine like terms.

2) 2y² – 4x + 3x – 5y² = (2y² – 5y²) + (-4x + 3x)	Regroup like terms.
= -3y² – 1x	Combine like terms.
= -3y² – x

3) 8xy² – x²y + 4x²y – 6xy²
= 8xy² – x²y + 4x²y – 6xy²	Or underline like terms without regrouping.
= 2xy² + 3x²y	Combine like terms.

4) 2(2m + 3n) + 3(m – 4n) = 4m + 6n + 3m – 12n	Distributive property.
= 7m – 6n	Combine like terms.

Removing Parentheses

If the sign preceding the parentheses is positive (+), do not change the sign of terms inside the parentheses, just remove the parentheses.

Example: (x – 5) = x – 5

If the sign preceding the parentheses is negative (-), remove the parentheses and the negative sign (in front of parentheses), and change the sign of each term inside the parentheses.

Example: – (x – 7) = –x + 7

Remove parentheses:

Algebraic expression	Remove parentheses	Example

(ax + b)	ax + b	(5x + 2) = 5x + 2
(ax – b)	ax – b	(9y – 4) = 9y – 4
– (ax + b)	-ax – b	– ( $\frac{3}{4}$ x + 7) = – $\frac{3}{4}$ x – 7
– (ax – b)	-ax + b	– (0.5b – 2.4) = -0.5b + 2.4

Example 1.1.2

1) 9x² + 7 – (2x² – 2) = 9x² + 7 – 2x² + 2	Remove parentheses.
= 7x² + 9	Combine like terms.

2) (-8y + 5z) – 4(y – 7z) = -8y + 5z – 4y + 28z	Remove parentheses.
= -12y + 33z	Combine like terms.

3) – (3a² + 4a – 4) + 3(4a² – 6a + 7)	Remove parentheses.
= – 3a² – 4a + 4 + 12a² – 18a + 21	Distributive property.
= 9a² – 22a + 25	Combine like terms.

4) -5(u² – 3u) + 3(2u – 4) – (5 – 3u + 4u²)	Distributive property.
= -5u² + 15u + 6u – 12 – 5 + 3u – 4u²	Remove parentheses.
= -9u² + 24u – 17	Combine like terms.

Multiplying and Dividing Algebraic Expressions

Multiplying a monomial and a polynomial:

Use the distributive property: a (b + c) = ab + ac
Multiply coefficients and add exponents with the same base. Apply a^m aⁿ = a^m+n

Example 1.1.3

1) 3x³ (5x² – 2x) = (3x³) (5x²) – (3x³) (2x)	Distributive property: a (b + c) = ab + ac
= (3 ∙ 5) (x³ x²) – (3 ∙ 2) (x³ x¹)	Regroup x = x¹
= 15 (x³⁺²) – 6 (x³⁺¹)	Multiply the coefficients & add the exponents.
= 15x⁵ – 6x⁴	a^m ∙ aⁿ = a^m+n

2) *5ab² (2a²b +ab² – a)*	Distribute.
= (5ab²) (2a²b) + (5ab²) (ab²) + (5ab²) (-a)	Multiply the coefficients and add exponents.
= (5 ∙ 2) (a¹⁺² b²⁺¹) + (5a¹⁺¹ b²⁺²) – (5a¹⁺¹b²)	b = b¹ , a = a¹
= 10a³b³ + 5a²b⁴ – 5a²b²	a^m ∙ aⁿ = a^m+n

Dividing a polynomial by a monomial:

Split the polynomial into several parts.
Divide a monomial by a monomial. Apply $\frac{a^m}{a^n}=a^{m-n}$ .

Example 1.1.4

$\frac{12x^2+4x-2}{4x}$

Steps Solution

Split the polynomial into three parts: $\frac{12x^2+4x-2}{4x}=\frac{12x^2}{4x}+\frac{4x}{4x}-\frac{2}{4x}$
Divide a monomial by a monomial: $=3x+1-\frac{1}{2x}$ $\frac{a^m}{a^n}=a^{m-n}$

The FOIL method: an easy way to find the product of two binomials (two terms).

(a + b) (c + d) = ac + ad + bc + bd F O I L	Example

F – First terms	first term × first term (a + b) (c + d)	(x + 5) (x + 4)
O – Outer terms	outside term × outside term (a + b) (c + d)	(x + 5) (x + 4)
I – Inner terms	inside term × inside term (a + b) (c + d)	(x + 5) (x + 4)
L – Last terms	last term × last term (a + b) (c + d)	(x + 5) (x + 4)