Advance Algebra 2 - Math Notes Unit 3 - Equivalent Forms
3.1.3 – Vocabulary for Expressions
A mathematical expression is a combination of numbers, variables, and operation symbols. Addition and subtraction separate
expressions into parts called terms. For example, 4x2 − 3x + 6 is an expression. It has three terms: 4x2, 3x, and 6.
The coefficients of the terms with variables are 4 and −3. 6 is called a constant term.
A single-variable polynomial is an expression that involves, at most, the operations of addition, subtraction, and
multiplication. Most of the polynomials you will work with can be written as expressions with terms of the following form:
(any real number)x(whole number)
For example, 4x2 – 3x1+ 6x0 is a polynomial, as is the simplified form, 4x2 – 3x+ 6. Also, since 6x0 = 6, 6 itself is a polynomial.
The function f(x) = 7x5+ 2.5x3− 1 x + 7 is a polynomial function.
2
A binomial is a polynomial with only two terms, for example, x3− 0.5x and 2x + 5.
1
The following expressions are not polynomials: 2x − 3, 2 , and √𝑥 − 2 .
𝑥 −2
1
An expression that can be written as the quotient of two polynomials is a rational expression. For example, 2 , is a rational
𝑥 −2
expression.
3.2.4 – Rewriting Rational Expressions
To simplify a rational expression, both the numerator and denominator must be written in factored form. Then look for factors that
make a "Giant One"(a form of the number 1) and simplify. Study Examples 1 and 2 below.
Example 1:
Example 2:
for x ≠ −4 or 3
for x ≠ −5 or
7
2
Just as you can multiply and divide fractions, you can multiply and divide rational expressions
Example 3: Multiply
𝑥 2 +6𝑥
(𝑥+6)2
∙
𝑥 2 +7𝑥+6
𝑥 2 −1
and simplify for x ≠ −6, –1, or 1.
After factoring, this expression becomes:
Since
(𝑥+6)
(𝑥+6)
Example 3: Divide
= 1 and
𝑥 2 −4𝑥−5
𝑥 2 −4𝑥+4
(𝑥+1)
(𝑥+1)
÷
Simplify:
=1
𝑥 2 −2𝑥−15
𝑥 2 +4𝑥−12
After multiplying, reorder the factors:
and simplify for x ≠ 2, 5, −3, or −6.
First, change to a multiplication expression:
𝑥 2 − 4𝑥 − 5 𝑥 2 + 4𝑥 − 12
∙
𝑥 2 − 4𝑥 + 4 𝑥 2 − 2𝑥 − 15
Then factor each expression:
(𝑥 − 5)(𝑥 + 1) (𝑥 − 2)(𝑥 + 6)
∙
(𝑥 − 2)(𝑥 − 2) (𝑥 − 5)(𝑥 + 3)
After multiplying, reorder the factors:
Since
(𝑥−5)
(𝑥−5)
= 1 and
(𝑥−2)
(𝑥−2)
= 1, simplify to get:
(𝑥 − 5) (𝑥 − 2) (𝑥 + 1) (𝑥 + 6)
∙
∙
∙
(𝑥 − 5) (𝑥 − 2) (𝑥 − 2) (𝑥 + 3)
**Note: From this point forward in the course, you may assume that all values of x that would make a denominator zero are excluded.
3.2.5 – Adding and Subtracting Rational Expressions
In order to add and subtract fractions, the fractions must have a common
denominator. One way to do this is to change each fraction so that the
denominator is the least common multiple of the denominators. For the
example right, the least common multiple of (x+ 3)(x + 2) and (x + 2) is
(x + 3)(x + 2).
The denominator of the first fraction already is the least common
multiple. To get a common denominator in the second fraction, multiply
(𝑥+3)
the fraction by
, a "Giant One" (a form of the number 1).
(𝑥+3)
Multiply the numerator and denominator of the second term.
Distribute the numerator, if necessary.
Add, factor, and simplify the result.