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Chapter 6

- Review of Factoring
- and Algebraic Fractions

Section 6.2 Factoring Common Factors and

Difference of Squares

Factoring is the reverse of multiplying. A

polynomial or a factor is called

_________________ if it contains no factors other

than 1 or -1.

THE FIRST STEP Factoring Out the Greatest

Common Monomial Factor

Factoring the Difference of Perfect Squares

Recall

Difference of Squares

Factoring the Difference of Perfect Squares

Factor Completely HINT Always check for a GCF

first!!

Factoring by Grouping(Consider grouping method

if polynomial has 4 terms)

- Always start by checking for a GCF of all 4

terms. After you factor out the GCF or if the

polynomial does not have a GCF other than 1,

check if the remaining 4-term polynomial can be

factored by grouping. - Determine if you can pair up the terms in such a

way that each pair has its own common factor. - If so, factor out the common factor from each

pair. - If the resulting terms have a common binomial

factor, factor it out.

Factor Completely

Factor Completely

Section 6.3 Factoring Trinomials

I. Factoring Trinomials in the Form

Recall

To factor a trinomial is to reverse the

multiplication process (UnFOIL)

Before you attempt to Un-FOIL?

1) Always factor out the GCF first, if

possible. 2) Write terms in descending order.

Now we begin?

3) Set up the binomial factors like this (x

)(x ) 4) List the factor pairs of

the LAST term If the LAST term is POSITIVE,

then the signs must be the same (both or both

-) If the LAST term is NEGATIVE, then the signs

must be different (one and one -). 5) Find the

pair whose sum is equal to the MIDDLE term 6)

Check by multiplying the binomials (FOIL)

Factor Completely

Factor Completely

Factoring Trinomials in the Form

The Trial Check Method

Before you attempt to Un-FOIL?

1) Always factor out the GCF first, if

possible. 2) Write terms in descending order.

Now we begin?

3) Set up the binomial factors like this ( x

)( x ) 4) List the factor

pairs of the FIRST term 5) List the factor pairs

of the LAST term 6) Sub in possible factor pairs

and try them by multiplying the binomials

(FOIL) until you find the winning combination

that is when OI MIDDLE term.

Factor completely

Factor completely

Factor completely

A General Strategy for Factoring Polynomials

Before you begin to factor, make sure the terms are written in descending order of the exponents on one of the variables. Rearrange the terms, if necessary. Factor out all common factors (GCF). If your leading term is negative, factor out -1. If an expression has two terms, check for the difference of two squares x2 - y2 (x y)(x - y) If an expression has three terms, attempt to factor it as a trinomial. If an expression has four terms, try factoring by grouping. Continue factoring until each individual factor is prime. You may need to use a factoring technique more than once. Check the results by multiplying the factors back out.

Section 6.5 Equivalent Fractions

The value of a fraction is unchanged if BOTH

numerator and denominator are multiplied or

divided by the same non-zero number.

Equivalent fractions

Equivalent fractions

An algebraic fraction is a ratio of two

polynomials. Some examples of algebraic

fractions are

Algebraic fractions are also called rational

expressions.

Simplifying Algebraic Fractions

A fraction is in its simplest form if the

numerator and denominator have no common factors

other than 1 or -1. (We say that the numerator

and denominator are relatively prime.)

We use terms like reduce, simplify, or put

into lowest terms.

Two simple steps for simplifying algebraic

fractions

FACTOR the numerator and the denominator. Divide out (cancel) the common FACTORS of the numerator and the denominator.

WARNING

Cancel only common factors. DO NOT CANCEL TERMS!

Example NEVER EVER NEVER do this!!!!!!!

Wrong! So very wrong!!

The correct way to simplify the rational

expression

- Here is the plan
- FACTOR the numerator and the denominator.
- Divide out any common FACTORS.

Simplest form.

Notice in this example

because the value of the denominator would be

0. ,

Simplify the rational expression

- FACTOR the numerator and the denominator.
- Divide out any common FACTORS.

A Special Case

The numerator and denominator are OPPOSITES.

Examples

Simplify each fraction.

Example

Simplify each fraction.