Transcript Document

Chapter 6
Review of Factoring
and
Algebraic Fractions
1
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.
2
THE FIRST STEP:
Factoring Out the Greatest Common Monomial Factor
1) 3x 2  42
2) 10a 2b 2  15ab3
3) 28 x 4 y 2  4 xy 3  4 y 2
4)  12m2 n  18mn
3
Factoring the Difference of Perfect Squares
(7 x  3)(7 x  3)
Recall:

Difference of Squares:
a  b  _______________________
2
2
4
Factoring the Difference of Perfect Squares
1) p  196
2) 4m  25n
3)  144  x 4 y 2
4) 9a 2  16
2
2
2
5
Factor Completely:
HINT: Always check for a GCF first!!
1)  324c  4cd
3
2
2) 16m  81n
4
4
6
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.
7
Factor Completely
1) 3m 2  15m  2mn  10n
2)  8wv  32v  88wv  352v
2
2
8
Factor Completely
3) p1R 2  p1r 2  p2 R 2  p2 r 2 ( fluid flow)
9
Section 6.3: Factoring Trinomials
2
x
 bx  c
I. Factoring Trinomials in the Form
Recall:
( x  5)( x  8)
 x  8 x  5 x  40
2
x 
3x
F
O+I
2
 40
L
To factor a trinomial is to reverse the multiplication process
(UnFOIL)
a2  7a 12   a
 a

10
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)
11
Factor Completely
1) m 2  14m  32
2) x 2  9 xy  18 y 2
3) 4a  77  a
4) r  14rt  49t
2
2
2
12
Factor Completely
5)  3q3  21q2  24q
13
Factoring Trinomials in the Form ax  bx  c
2
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;
14
that is when O+I =MIDDLE term.
Factor completely
1) 5 x  31x  28
2
2) 4 x  13  28 x
2
15
Factor completely
3) 2 x  3x  15
2
4) 12 x  x  20
2
16
Factor completely
5) 12 g  21g 2  9 g 3
17
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.
1. Factor out all common factors (GCF). If your leading term is
negative, factor out -1.
2. If an expression has two terms, check for the difference of two
squares: x2 - y2 = (x + y)(x - y)
3. If an expression has three terms, attempt to factor it as a trinomial.
4. If an expression has four terms, try factoring by grouping.
5. Continue factoring until each individual factor is prime. You may
need to use a factoring technique more than once.
6. Check the results by multiplying the factors back out.
18
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.
5
5  3 15


12 12  3 36
Equivalent fractions
18 18  6 3


24 24  6 4
Equivalent fractions
19
An algebraic fraction is a ratio of two polynomials.
Some examples of algebraic fractions are:
2
2x y
5 xy
3
,
3m  1
6
,
and
2
x
9m  1
Algebraic fractions are also called rational expressions.
20
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:
1. FACTOR the numerator and the denominator.
2. Divide out (cancel) the common FACTORS of
the numerator and the denominator.
21
WARNING:
Cancel only common factors.
DO NOT CANCEL TERMS!
Example: NEVER EVER NEVER do this!!!!!!!
1
2
3
x2  9
x 9

x 2  8 x  15 x 2  8 x  15
1
Wrong! So very wrong!!
5
22
The correct way to simplify the rational expression
Here is the plan:
1. FACTOR the numerator and the denominator.
2. Divide out any common FACTORS.
x2  9
x  3 x  3


x 2  8 x  15
 x  3 x  5
x  3 x  3


 x  3 x  5

 x  3
 x  5
Simplest form.
Notice in this example x  3 and x  5 because the value of
the denominator would be 0.
,
23
Simplify the rational expression
x 2  x  30
x 2  25
1. FACTOR the numerator and the denominator.
2. Divide out any common FACTORS.
x 2  x  30
x 2  25
24
A Special Case
The numerator and denominator are OPPOSITES.
1)
3 1 3

 1
3
3
b  a 1 b  a  1 a  b 
2)


 1
a b
a b
a b
3)
x4
x4
1


 1
4  x 1 x  4  1
25
Examples
Simplify each fraction.
7 a 2  21a
1)
14a
x2  4
2)
2 x 2  3x  2
26
Example
Simplify each fraction.
3a 2  13a  10
3)
5  4a  a 2
27