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Factoring Polynomials

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Title: Factoring Polynomials


1
Factoring Polynomials
  • 1 The Greatest Common Factor and Factoring By
    Grouping
  • 2 Factoring Trinomials Whose Leading Coefficient
    is 1
  • 3 Factoring Trinomials Whose Leading Coefficient
    is Not 1
  • 4 Factoring Special Forms
  • 5 A General Factoring Strategy
  • 6 Solving Quadratic Equations by Factoring

2
The Greatest Common Factor and Factoring By
Grouping
  • Objectives Factor monomials
  • Find the greatest common factor
  • Factor out the greatest common factor of a
    polynomial
  • Factor by grouping

3
Overview Factoring out the Greatest Common
Factor (GCF)
  • Background

Examples 2 36 Factor 6 2 3
Notes In Arithmetic 2 and 3 are factors of 6
because they multiply to 6 Means change it back
to a multiplication problem Check by multiplying
the factors did you get back the original
number?
4
Overview Factoring out the Greatest Common
Factor (GCF)
  • Background

Examples Factor 12 2 6, or 3 4 or 2 2 3
Notes Means change it back to a multiplication
problem Factored-but not completely. 6 is not
prime, replace it with 2 34 is not prime,
replace it with 2 2 This is the prime
factorization-factored completely.
5
Overview Factoring out the Greatest Common
Factor (GCF)
  • Background Continued

Notes In Algebra This is considered a
Multiplication Problem (Last operation, if you
had a value for x, would be to multiply) (order
of operations Do inside ( ) first, then all mult
div from left to right) It is ONE term, with
TWO factors 2x3 and (x2 3)
Examples 2x3(x2 3)
6
Overview Factoring out the Greatest Common
Factor (GCF)
  • Background Continued

Examples 2x3(x2 3) 2x3 x2 2x3
3 2x5 6x3
Notes To Multiply Distribute 2x3 over x2 3
(Could be done mentally) Now its an addition
problem. Why? How many terms? 2x3 is a common
factor of the terms
Term 1 Term 2
7
Overview Factoring out the Greatest Common
Factor (GCF)
  • We now try to go the other way-Factor.

Notes See that it is an addition problem with 2
terms, and we want to re-write it as a
multiplication problem. Pull out the GCF
2x3 Now how many terms?
Examples Factor 2x5 6x3 2xxxxx 2
3xxx 2x3 x2 2x3 3 2x3(x2 3)
8
Overview Factoring out the Greatest Common
Factor (GCF)
  • In General
  • We use the distributive property to multiply
  • a(b c) ab ac
  • We reverse the distributive property to factor
  • ab ac

9
Objective 1 Factoring monomials
  • Factoring a monomial means finding two monomials
    whose product gives the original monomial.
  • For example 30x2 can be factored in a number of
    different ways
  • (5x)(6x) (3x)(10x) (2x2)(15)

10
Objective 2 Finding the Greatest Common
Factor
  • How do we find the greatest common factor (GCF)
    to pull out?
  • Find the largest integer that divides the
    coefficients.
  • List each variable that is common to all the
    monomials, use the lowest power of that variable
    from the monomials.
  • 3. The GCF of the monomials is the product of the
    coefficient determined in step 1 and the variable
    factor(s) determined in step 2.

11
Objective 3 Factoring out the Greatest
Common Factor
  • Factoring a Monomial from a Polynomial
  • Determine the greatest common factor of all terms
    in the polynomial.
  • Express each term as the product of the GCF and
    its other factor.
  • 3. Use the Distributive Property to factor out
    the GCF.

Examples Factor the following a) 16 y5 - 12
y4 8y3 b) -3 y2 - 15 y - 6
12
Objective 4 Factoring by Grouping
  • 1. Group terms that have a common monomial
    factor. There will usually be two groups.
    Sometimes the terms must be rearranged.
  • 2. Factor out the common monomial factor from
    each group.
  • 3. Factor out the remaining binomial factor (if
    one exists)

Examples Factor the following by
Grouping a) x3 - 3x2 2x - 6 b) 4bx - 3b - 20
x 15
13
Factoring Trinomials Whose Leading Coefficient is
1
  • Objective
  • Factoring trinomials of the form x2 bx c

14
Objective 1 Factoring Trinomials of the form
1x2 bx c
  • 1. Enter x as the first term of each factor
  • x2 bx c ( x ) ( x )
  • 2. To determine the second term of each factor
  • a) Find all pairs of integers whose product is
    c, the third term of the trinomial.
  • b) Choose the pair whose sum is b, the
    coefficient of the middle term of the trinomial.

15
Objective 1 Factoring Trinomials of the form
1x2 bx c
  • b) Choose the pair whose sum is b, the
    coefficient of the middle term of the trinomial.
  • c) If mn c and m n b, then m and n are the
    desired integers, and
  • x2 bx c ( x m ) ( x n )
  • 3. If there are no such integers, the trinomial
    cannot be factored and is called prime.

16
Application problem
  • Consider a person standing at the edge of a cliff
    who throws a rock upward with an initial speed of
    64 feet per second. His hand at the time of
    release is 80 feet above the water. After t
    seconds, the height h of the rock above the water
    is described by the model
  • h -16t 2 64t 80
  • Factor this polynomial completely. Begin by
    factoring 16 from each term.

80 ft
17
Application problem continued
  • h -16t 2 64t 80
  • Factor this polynomial completely. Begin by
    factoring 16 from each term.
  • How can we determine how long it takes for the
    rock to enter the water?

80 ft
18
Objective 1 Factoring Trinomials of the form
1x2 bx c Revisited
  • Find two integers m and n whose product is c and
    whose sum is b. If mn c and m n b, then
  • x2 bx c ( x m ) ( x n )
  • 1. If c gt 0 m and n must be the same sign!!
  • If b gt 0 and c gt 0, m and n must be positive.
  • If b lt 0 and c gt 0, m and n must be negative.
  • 2. If c lt 0, m and n must have opposite signs.

19
Objective 1 Factoring Trinomials of the form
1x2 bx c Revisited
  • Try a problem from your text. Remember to first
    remove any GCF!
  • 1. If c gt 0 m and n must be the same sign!!
  • If b gt 0 and c gt 0, m and n must be positive.
  • If b lt 0 and c gt 0, m and n must be negative.
  • 2. If c lt 0, m and n must have opposite signs.

20
Objective 1 Factoring Trinomials of the form
1x2 bx c
  • Example A prime polynomial that wont factor.
  • Example Factoring a Trinomial in Two Variables
  • Example Factoring Completely.

21
Factoring Trinomials Whose Leading Coefficient is
Not 1
  • Objectives
  • 1. Factor trinomials by trial and error
  • 2. Factor trinomials by grouping
  • 3. Factor trinomials by the box method (Not
    in Text)
  • I will do the objectives in reverse!

22
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
Background
  • 1a. Multiply (2x 3) (3x 2) using the box

Find the product of both diagonals. Where does
the middle term of the answer come from?
23
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1b. Factor 6x2 13x 6 using the box

First check all three terms for a GCF!
Find the product of the one diagonal.
24
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1b. Factor 6x2 13x 6 using the box

We have 2 empty boxes with 2 clues!
25
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1b. Factor 6x2 13x 6 using the box

We have 2 empty boxes with 2 clues! Clue 1 The
PRODUCT of the coefficients must be_____ Clue 2
The SUM of the coefficients must be _____
26
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1b. Factor 6x2 13x 6 using the box

Product of 36 Test the sum(Signs same!)
(Stop if adds to13 or opp)(Divide to get 2nd
factor) 1 36 1 36 37 no
2 18 3 ?
27
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1b. Factor 6x2 13x 6 using the box

Product of 36 Test the sum(Signs same!)
(Stop if adds to13 or opp)(Divide to get 2nd
factor) 9 4 9 4 13 YES!
So 9x and 4x go into the empty boxes!
Either one in either box.
28
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1b. Factor 6x2 13x 6 using the box

4x
9x
OR
9x
4x
29
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1b. Factor 6x2 13x 6 using the box

Pull out only ONE greatest common factor (GCF)
from any horizontal row or vertical column
30
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1b. Factor 6x2 13x 6 using the box

Remember, the 2 monomials outside the box
multiply to give the monomial inside.
31
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1b. Factor 6x2 13x 6 using the box

32
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1b. Factor 6x2 13x 6 using the box

The factors are (3x 2) (2x 3) Always check
by multiplying! Note how the boxes fill in
again, to help you with the factoring process.
33
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1a. Factor 10x2 - 17x 3 using the box

Find the product of the one diagonal.
34
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1a. Factor 10x2 - 17x 3 using the box

Product of 30 Test the sum (Signs same!
Why?) (Stop if adds to -17 or opp)(divide to get
2nd factor) 1 30 1 30 31
no 2 15 2 15 17 STOP!
Take opposite of BOTH
-2 -15 -17
35
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1a. Factor 10x2 - 17x 3 using the box

36
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1c. Factor 12x2 7x 12 using the box

First check all three terms for a GCF
We have 2 empty boxes with 2 clues!
37
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1c. Factor 12x2 7x 12 using the box

Product of -144 Test the sum (Signs -
Why?) (Stop if adds to 7 or opp)(divide to
get 2nd factor) 1 -144 1 -144
-143 no (different
signs Subtract!) 2 - 72 2 -72
-70 no
If youre alert and see your sums are negative
and your middle term is positive, switch and make
the larger factor and smaller -. Or, when you
add and get 7, switch then.
38
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1c. Factor 12x2 7x 12 using the box

Product of -144 Test the sum (Signs -
Why?) (Stop if adds to 7 or opp)(divide to
get 2nd factor) -3 ? -4
? Remember to use your calculator if need be to
divide -144/ (-3) to get the other factor.
If it doesnt divide evenly, its not a factor!

39
Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)
  • 1c. Factor 12x2 7x 12 using the box

Pull out only ONE GCF! Then find all the missing
? Watch your signs!
40
Objective 2 Factoring Trinomials Whose Leading
Coefficient is Not 1 by the factor by grouping
method
  • 1c. Factor 12x2 7x 12 using the factoring
    by grouping method
  • 0. First check all three terms for a GCF!
  • There isnt a GCF. Next, to factor by grouping
    you need at least 4 terms. We only have 3.
  • The goal is to recreate the 4 terms. Remember
    the 2 like terms (inners and outers) were
    added together. We want to reverse the process.

41
Objective 2 Factoring Trinomials Whose Leading
Coefficient is Not 1 by the factor by grouping
method
  • 1c. Factor 12x2 7x 12 using the factoring
    by grouping method
  • Factoring ax2bxc Using Grouping a?1
  • Multiply the leading coefficient, a, and the
    constant c (12 -12 -144)
  • List the factors of ac, Find the ones that add to
    b
  • 1 -144 Sum 1 -144 -143
    no
  • 2 - 72 Sum 2 -72 -70
    no

0. First check all three terms for a GCF!
42
Objective 2 Factoring Trinomials Whose Leading
Coefficient is Not 1 by the factor by grouping
method
  • 1c. Factor 12x2 7x 12 using the factoring
    by grouping method

-3 ? -4 ? -5 ?
-6 ? -7 ? -8 ? -9 ?
(Find the ?s by division) 3. Rewrite the
middle term as a sum or difference using the
factors from step 2. 7x -9x 16x 4.
Factor by grouping 12x2 - 9x 16x - 12
43
Objective 2 Factoring Trinomials Whose Leading
Coefficient is Not 1 by the factor by grouping
method
  • 1c. Factor 12x2 7x 12 using the factoring
    by grouping method
  • Now, Factor by grouping 12x2 - 9x 16x 12
  • (12x2 - 9x) (16x 12)
  • 3x(4x 3) 4(4x 3)
  • (4x 3) (3x 4)

Regroup Factor out GCF from each group Factor out
the common Binomial Factor
44
Objective 1 Factoring Trinomials Whose Leading
Coefficient is Not 1 by Trial and Error (Guess by
golly!)
  • Trial and Error works best with small or prime
    numbers-there are fewer combinations!
  • Example 1-
  • Factor 3x2 20x 28 by trial and error
  • ( ) ( )
  • Step 1 The first term had to come from 3x x!
  • Step 2 The last term has a few more
    possibilities

45
7.3 Objective 1 Factoring Trinomials Whose
Leading Coefficient is Not 1 by Trial and Error
(Guess by golly!)
  • Example
  • Factor 3x2 20x 28 by trial and error
  • (3x ) (x )
  • Step 2 The last term has a few more
    possibilities.
  • Its positive 28, so the 2 factors must have the
    same sign both positive or both negative.
  • How do we decide which?

46
7.3 Objective 1 Factoring Trinomials Whose
Leading Coefficient is Not 1 by Trial and Error
(Guess by golly!)
  • Example
  • Factor 3x2 20x 28 by trial and error
  • (3x ) (x )
  • Step 2 The last term has a few more
    possibilities.
  • Its positive 28, so the 2 factors must have the
    same sign both positive or both negative.
  • How do we decide which?
  • The middle term is 20x, is negative so
    both factors of 28 must be negative.

47
7.3 Objective 1 Factoring Trinomials Whose
Leading Coefficient is Not 1 by Trial and Error
(Guess by golly!)
  • Example
  • Factor 3x2 20x 28 by trial and error
  • (3x ) (x )
  • Step 2 We need the factors of 28, but we cant
    just take the sum and try to get 20. One of the
    factors is multiplied by 3 first!
  • The possible factor pairs are -1(-28), -2(-14)
    and 4(-7)

48
7.3 Objective 1 Factoring Trinomials Whose
Leading Coefficient is Not 1 by Trial and Error
(Guess by golly!)
  • Example
  • Factor 3x2 20x 28 by trial and error
  • (3x 1) (x 28) or (3x 28)
    (x 1)
  • Step 3
  • The Inners Outers must add to 20x

49
7.3 Objective 1 Factoring Trinomials Whose
Leading Coefficient is Not 1 by Trial and Error
(Guess by golly!)
  • Example 1
  • Factor 3x2 20x 28 by trial and error
  • (3x 2) (x 14) or (3x 14)
    (x 2)
  • Step 3
  • Bingo! -14x -6x -20x
  • (3x 14) (x 2) is it

50
7.4 Factoring Special Forms
  • Objectives
  • Factor the difference of two squares.
  • Factor perfect square trinomials.
  • Factor the sum and difference of two cubes.
    (Optional)

51
7.4 Objective 1 Factoring the Difference of
Two Squares.
  • Recall the special products from Chapter 6
  • Multiply (A B) (A B)
  • A2
    AB AB B2
  • The Middle term drops out! A2 B2
  • The product is called the difference of two
    squares
  • So, to factor A2 B2 , recognize that you are
    subtracting (the difference) the squares of two
    terms.

52
7.4 Objective 1 Factoring the Difference of
Two Squares.
  • So, to factor A2 B2 , recognize that you are
    subtracting (the difference) the squares of two
    terms.
  • P421 The Difference of Two Squares
  • If A and B are real numbers, variables or
    algebraic expressions, then
  • A2 B2 (A B) (A B)
  • In words The difference of the squares of two
    terms is factored as the product of the sum and
    the difference of those terms.

53
7.4 Objective 1 Factoring the Difference of
Two Squares.
  • Finding the two terms is sometimes the challenge.
  • Factor 36x2 25
  • May help to rewrite as
  • (6x)2 52
  • Then factor
  • 36x2 25 (6x 5) (6x 5)

54
7.4 Objective 1 Factoring the Difference of
Two Squares.
  • Finding the two terms is sometimes the challenge.
  • Factor x2 1 (and the simple ones are often
    not so obvious! 12 1

Factor 9 16 x10 (Realize 32 9 and
(4x5) 2 16 x10)
55
7.4 Objective 2 Factoring Completely
Factoring Perfect Square Trinomials
  • Example 3 Factoring Out the GCF and then
    Factoring the Difference of Two Squares
  • Example 4 A Repeated Factorization
  • Example 5 Factoring Perfect Square Trinomials
    (Can also be done by trial and error.)

56
Factoring Special Forms
Difference of Two Squares If a and b are real
numbers, variables, or algebraic expressions,
then A2 - B2 (A B) (A - B) Perfect Square
Trinomials If a and b are real numbers,
variables, or algebraic expressions,
1. A 2 2 A B B2
(A B)2
2. A 2 - 2 A B B2 (A - B)2
57
7.4 Objective 3 Factoring the Sum and
Difference of Two Cubes. (optional)
  • We know the difference of 2 squares will factor
  • A2 B2 (A B)(A B)


58
7.4 Objective 3 Factoring the Sum and
Difference of Two Cubes. (optional)
  • The Sum of 2 Squares does not factor with real
    numbers
  • A2 B2 Cannot be factored!

59
7.5 A General Factoring Strategy
  • Objectives
  • Recognize the appropriate method for factoring a
    polynomial.
  • Use a general strategy for factoring polynomials.

60
7.5 Objective 1 A Strategy for Factoring
Polynomials
  • If there is a common factor, factor out the GCF.
  • Determine the number of terms in the polynomial
    and try factoring as follows.
  • If there are 2 terms, can the binomial be
    factored by one of the following special forms?
  • Difference of two squares A2 - B2 (A
    B) (A - B)
  • Sum of two cubes A3 B3 (A B) (A2 - AB
    B2)
  • Difference of two cubes A3 - B3 (A - B) (A2
    AB B2)

61
7.5 Objective 1 A Strategy for Factoring
Polynomials
  • If there are 3 terms
  • Look for special trinomial forms
  • Perfect Square (Sum) A2 2AB B2 (A B)2
  • Perfect Square (Difference) A2 - 2AB B2
    (A - B)2
  • If the trinomial is not a perfect square
    trinomial, try factoring by trial and error, by
    ac-split middlegrouping, or by box
  • If there are four or more terms, try factoring by
    grouping.

62
7.5 Objective 1 A Strategy for Factoring
Polynomials
  • Check to see if any factors with more than one
    term in the factored polynomial can be factored
    further. If so, factor completely.
  • Check by multiplying

63
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64
7.6 Solving Quadratic Equations by Factoring
  • Objectives
  • Use the zero-product principle.
  • Solve quadratic equations by factoring.
  • Solve problems using quadratic equations.

65
7.6 Solving Quadratic Equations by Factoring
  • The alligator, an endangered species, was the
    subject of a protection program at Floridas
    Everglades national Park.

66
7.6 Solving Quadratic Equations by Factoring
  • Park rangers used the formula
  • P -10x2 475x 3500to estimate the alligator
    population, P, after x years of the protection
    program.
  • Their goal was to bring the population up to 7250

67
7.6 Solving Quadratic Equations by Factoring
  • To find out how long this would take to occur, we
    would need to solve the following equation for x.
  • 7250 -10x2 475x 3500
  • How does this differ from a linear equation?

68
7.6 Solving Quadratic Equations by Factoring
  • 7250 -10x2 475x 3500
  • Solving still means finding the numbers that make
    it true.
  • Why cant we get x to occur only once?
  • In this section, we will use factoring to solve
    equations in the form ax2 bx c 0
  • We also look at applications of these equations.

69
7.6 Definition of a Quadratic Equation
  • Equations that can be written in the form ax2
    bx c 0
  • are called Quadratic Equations in x.
  • a, b and c are real numbers, and a ? 0
  • A quadratic equation in x is also called a
    second-degree polynomial equation in x

70
7.6 Definition of a Quadratic Equation
  • ax2 bx c 0
  • For the quadratic equation
  • x2 - 7 x 10 0
  • Find a, b and c

71
7.6 Objective 1 The Zero- Product Principle
  • Factor the left side of the quadratic equation
  • x2 - 7 x 10 0
  • (x 5)(x 2) 0

Now we have a product that equals 0
72
7.6 Objective 1 The Zero-Product Principle
  • x2 - 7 x 10 0
  • (x 5)(x 2) 0
  • If a quadratic equation has zero on one side and
    a factored expression (multiplication!) on the
    other, it can be solved using the Zero-product
    principle

73
7.6 Objective 1 Using the Zero-Product
Principle
  • Zero-Product Principle
  • If the product of two algebraic expressions is
    zero, then at least one of the factors is equal
    to zero.
  • If AB 0, the A0 or B0

74
7.6 Objective 1 The Zero-Product Principle
  • x2 - 7 x 10 0
  • (x 5)(x 2) 0
  • According to the zero-product principle, this
    product can be zero only if at least one of the
    factors 0
  • Set each individual factor 0 and solve each
    resulting equation for x.

75
Objective 1 The Zero-Product Principle
  • x2 - 7 x 10 0
  • (x 5)(x 2) 0
  • Set each individual factor 0 and solve each
    resulting equation for x.
  • x 5 0 or x 2 0
  • x 5 x 2

76
Objective 1 The Zero-Product Principle to
Solve Quadratics
  • x2 - 7 x 10 0
  • (x 5)(x 2) 0
  • x 5 0 or x 2 0
  • x 5 x 2
  • Check each proposed solution in the original
    equation.

77
Objective 1 The Zero-Product Principle
78
Objective 2 Solve quadratic equations by
factoring
  • If necessary, rewrite the equation in the form
    ax2 bx c 0, moving all terms to one side,
    thereby obtaining zero on the other side.
  • Factor
  • Apply the zero-product principle, setting each
    factor equal to zero.
  • Solve the equations in step 3.
  • Check the solutions in the original equation.

79
Objective 2 Solve quadratic equations by
factoring
  • Now try
  • Its not 0!!
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