Factoring Polynomials

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Chapter 5

• Factoring Polynomials

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5-1 Factoring Integers

• Factors - integers that are multiplied together
  to produce a product.
• 4 x5 20

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2,3,5,7,11,13,17,19,23,29

• Prime number - is an integer greater than 1 that
  has no positive integral factor other than itself
  and 1.

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PRIME FACTORIZATION

• Prime factorization of 36
• 36 2 x 18
• 2 x 2 x 9
• 2 x 2 x 3 x 3
• 22 x 32

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GREATEST COMMON FACTOR

• The greatest integer that is a factor of all the
  given integers.

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GREATEST COMMON FACTOR

• Find the GCF of 25 and 100
• 25 5 x 5
• 100 2 x 2 x 5 x 5
• GCF 5 x 5 25

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• 5-2 Dividing Monomials

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Property of Quotients

• If a, b, c and d are real numbers, then
• ac a c
• bd b d

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Simplifying Fractions

• If b, c and d are real numbers, then
• bc c
• bd d

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Rule of Exponents for Division

• If a is a nonzero real number and m and n are
  positive integers, and m gt n, then am am-n
• an

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Rule of Exponents for Division

• If a is a nonzero real number and m and n are
  positive integers, and
• n gt m then am 1
• an an-m

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Rule of Exponents for Division

• If a is a nonzero real number and m and n are
  positive integers, and
• m n then am 1
• an

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GREATEST COMMON FACTOR

• The greatest common factor of two or more
  monomials is the common factor with the greatest
  coefficient and the greatest degree in each
  variable.

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GREATEST COMMON FACTOR

• Find the GCF of 25x4y and 50x2y5
• GCF 25x2y

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• 5-3
• Monomial Factors of Polynomials

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Dividing a Polynomial by a Monomial

• Divide each term of the polynomial by the
  monomial and add the results.

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Dividing Polynomials by Monomials

• 5m 35 m 7
• 5
• 7x2 14x x 2
• 7x

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Factoring a Polynomial

• To factor
• Find the GCF
• Divide each term by the GCF
• Write the product

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Examples

• 5x2 10x
• 4x5 6x3 14x
• 8a2bc2 12ab2c2

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5-4 Multiplying Binomials Mentally

• When multiplying two binomials both terms of each
  binomial must be multiplied by the other two
  terms

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Binomial

• A polynomial that has two terms
• 2x 3 4x 3y
• 3xy 14 613 39z

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Trinomial

• A polynomial that has three terms
• 2x2 3x 1
• 14 32z 3x
• mn m2 n2

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Multiplying binomials

• Using the F.O.I.L method helps you remember the
  steps when multiplying

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F.O.I.L. Method

• F multiply First terms
• O multiply Outer terms
• I multiply Inner terms
• L multiply Last terms
• Add all terms to get product

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Example (2a b)(3a 5b)

• F 2a 3a
• O 2a 5b
• I (-b) ? 3a
• L - (-b) ? 5b
• 6a2 10ab 3ab 5b2
• 6a2 7ab 5b2

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Example (x 6)(x 4)

• F x ? x
• O x ? 4
• I 6 ? x
• L 6 ? 4
• x2 4x 6x 24
• x2 10x 24

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• Section 5-5
• Difference of Two Squares

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Multiplying

• (x 3) (x - 3) ?
• (y - 2)(y 2) ?
• (s 6)(s 6) ?

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Factoring Pattern

• a2 b2 (a b) (a b)

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FACTOR

• x2 - 49 ?
• 16 y2 ?
• 81t2 25x6 ?

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• 5-6 Squares of Binomials

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Examples - Multiply

• (x 3)2 ?
• (y - 2)2 ?
• (s 6)2 ?

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Factoring Patterns

• (a b)2 a2 2ab b2
• (a - b)2 a2 - 2ab b2
• Also known as Perfect square trinomials

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Examples Factor

1. 4x2 20x 25
2. 64u2 72uv 81v2
3. 9m2 12m 4
4. 25y2 5y 1

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• 5-7
• Factoring Pattern for x2 bx c, c positive

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Example

• x2 8x 15
• Middle term is the sum of 3 and 5
• Last term is the product of 3 and 5

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Example

• y2 14y 40
• Middle term is the sum of 10 and 4
• Last term is the product of 10 and 4

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Example

• y2 11y 18
• Middle term is the sum of -2 and -9
• Last term is the product of -2 and -9

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Factor

1. m2 3m 5
2. k2 9k 20
3. y2 9y 8

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• 5-8
• Factoring Pattern for x2 bx c, c negative

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• x2 - x - 20
• Middle term is the sum of 4 and -5
• Last term is the product of 4 and -5

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Example

• y2 6y - 40
• Middle term is the sum of 10 and -4
• Last term is the product of 10 and -4

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Example

• y2 7y - 18
• Middle term is the sum of 2 and -9
• Last term is the product of 2 and -9

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Factor

1. x2 4kx 12k2
2. p2 32p 33
3. a2 3ab 18b2

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5-9 Factoring Pattern for ax2 bx c

• List the factors of ax2
• List the factors of c
• Test the possibilities to see which produces the
  correct middle term

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Examples

• 2x2 7x 9
• 14x2 - 17x 5
• 10 11x 6x2
• 5a2 ab 22b2

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5 -10 Factor by Grouping

• Factor each polynomial by grouping terms that
  have a common factor
• Then factor out the common factor and write the
  polynomial as a product of two factors

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Examples

• xy xz 3y 3z
• 3xy 4 6x 2y
• xy 3y 2x 6
• ab 2b ac 2c
• 9p2 t2 4ts 4s2

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5 -11 Using Several Methods of Factoring

• A polynomial is factored completely when it is
  expressed as the product of a monomial and one or
  more prime polynomials.

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Guidelines for Factoring Completely

• Factor out the greatest monomial factor first
• Factor the remaining polynomial

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Guidelines for Factoring Completely

• Make sure that each binomial or trinomial factor
  is prime.

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Example - Factor

• -4n4 40n3 100n2
• 5a3b2 3a4b 2a2b3
• a2bc - 4bc a2b - 4b

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5 -12 Solving Equations by Factoring

• Zero-Product Property
• For all real numbers a and b
• ab 0
• if and only if
• a 0 or b 0

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Examples

• 1. (x 2) (x 5) 0
• 2. 5n(n 3)(n 4) 0
• 3. 2x2 5x 12
• 4. 18y3 8y 24y2 0

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5 -13 Using Factoring to Solve Word Problems

• Suppose Mike bought 36 feet of wire to make a
  rectangular pen for his pet. If he wants the
  area to be 80 ft2, what are the dimensions he
  could use?

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Solution

• Let x Length, then Width (36 2x)/2 18 x
• 80 18x x2
• x2 18x 80 0
• (x 10) (x-8) 0
• 8, 10

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END

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