1.1. Introduction to Algebra — Mathematics for Public and Occupational Health Professionals

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75 1.1. Introduction to Algebra — Mathematics for Public and Occupational Health Professionals

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Introduction

Review of basic algebraic terms:

Algebraic term	Description	Example

Algebraic expression

A mathematical phrase that contains numbers, variables (letters), and arithmetic operations (+, – , ×, ÷, etc.).

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.)

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: 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.

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

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^maⁿ = a^m+n

Dividing a polynomial by a monomial:

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

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)

FOIL method	Example

(a + b) (c + d) = ac + ad + bc + bd	(x + 5) (x + 4) = x ∙ x + x ∙ 4 + 5x + 5 ∙ 4 = x² + 9x + 20
F O I L	F O I L

Multiplying binomials (2 terms × 2 terms):

F O I L

The FOIL method.

$=10x^2-12x+15x-18$

aⁿ a^m = a^{n+ m}

$=10x^2+2x-18$

Combine like terms.

2) $(3r - t)(5r+t^2)=3r \cdot 5r+3r \cdot t^2-t \cdot 5r-t \cdot t^2$	FOIL
$=15r^2+3rt^2-5rt-t^3$	aⁿ a^m = a^n+m
$=18r^2+-5rt-t^3$	Combine like terms.

3) $(xy^2+y)(2x^2y+x)=xy^2 \cdot 2x^2y+xy^2 \cdot x+y \cdot 2x^2y+yx$	FOIL
$=2x^3y^3+x^2y^2+2x^2y^2+xy$	aⁿ a^m = a^n+m
$=2x^3y^3+3x^2y^2+xy$	Combine like terms.