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?
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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.
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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)
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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
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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)
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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

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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)

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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.

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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
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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
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Factoring Trinomials Whose Leading Coefficient is
1

• Objective
• Factoring trinomials of the form x2 bx c

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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.

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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.

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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
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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
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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.

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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.

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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.

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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!

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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?
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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.
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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!
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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 _____
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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 ?
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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.
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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
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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
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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.
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Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)

• 1b. Factor 6x2 13x 6 using the box

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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.
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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.
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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
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Factoring Trinomials Whose Leading Coefficient is
Not 1 by the box method (Not in Text)

• 1a. Factor 10x2 - 17x 3 using the box

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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!
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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.
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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!

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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!
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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.

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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!
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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
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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
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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

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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?

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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.

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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)

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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

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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

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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)

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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.

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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.

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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)

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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)
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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.)

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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
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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)

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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!

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7.5 A General Factoring Strategy

• Objectives
• Recognize the appropriate method for factoring a
  polynomial.
• Use a general strategy for factoring polynomials.

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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)

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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.

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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

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7.6 Solving Quadratic Equations by Factoring

• Objectives
• Use the zero-product principle.
• Solve quadratic equations by factoring.
• Solve problems using quadratic equations.

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7.6 Solving Quadratic Equations by Factoring

• The alligator, an endangered species, was the
  subject of a protection program at Floridas
  Everglades national Park.

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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

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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?

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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.

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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

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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

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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
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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

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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

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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.

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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

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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.

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Objective 1 The Zero-Product Principle
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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.

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Objective 2 Solve quadratic equations by
factoring

• Now try
• Its not 0!!