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Review of Factoring

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Chapter 6 Review of Factoring and Algebraic Fractions * MAT 105 FALL 2008 MAT 105 FALL 2008 MAT 105 FALL 2008 MAT 105 FALL 2008 MAT 105 FALL 2008 MAT 105 FALL 2008 ... – PowerPoint PPT presentation

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Title: Review of Factoring


1
Chapter 6
  • Review of Factoring
  • and Algebraic Fractions

2
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.
3
THE FIRST STEP Factoring Out the Greatest
Common Monomial Factor
4
Factoring the Difference of Perfect Squares
Recall
Difference of Squares
5
Factoring the Difference of Perfect Squares
6
Factor Completely HINT Always check for a GCF
first!!
7
Factoring by Grouping(Consider grouping method
if polynomial has 4 terms)
  1. 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.
  2. Determine if you can pair up the terms in such a
    way that each pair has its own common factor.
  3. If so, factor out the common factor from each
    pair.
  4. If the resulting terms have a common binomial
    factor, factor it out.

8
Factor Completely
9
Factor Completely
10
Section 6.3 Factoring Trinomials
I. Factoring Trinomials in the Form
Recall
To factor a trinomial is to reverse the
multiplication process (UnFOIL)
11
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)
12
Factor Completely
13
Factor Completely
14
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.
15
Factor completely
16
Factor completely
17
Factor completely
18
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.
19
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
20
An algebraic fraction is a ratio of two
polynomials. Some examples of algebraic
fractions are
Algebraic fractions are also called rational
expressions.
21
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.
22
WARNING
Cancel only common factors. DO NOT CANCEL TERMS!
Example NEVER EVER NEVER do this!!!!!!!
Wrong! So very wrong!!
23
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. ,

24
Simplify the rational expression
  1. FACTOR the numerator and the denominator.
  2. Divide out any common FACTORS.

25
A Special Case
The numerator and denominator are OPPOSITES.
26
Examples
Simplify each fraction.
27
Example
Simplify each fraction.
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