Section 5'5 Factoring: A General Review

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Section 5.5Factoring A General Review

• Strategy for factoring polynomials

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Strategy for Factoring a Polynomial

• GCF
• Two terms check for the difference of squares
• Three terms check for perfect square trinomial
• Inspection
• mn Grouping
• More than three terms Grouping
• Can you factor further?

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Step 1 GCF

• If the polynomial has a greatest common factor
  other than 1, then factor out the greatest common
  factor.
• 5x2 10x ?
• 5x(x 2)

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Step 1 GCF

• 14x 21
• 9x 12y
• 2x2 6x 4
• 5ab2 10a2b2 15a2b

• 7(2x 3)
• 3(3x 4y)
• 2(x2 3x 2)
• 5ab(b 2ab 3a)

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Step 1 GCF

• 5a 7a
• a(5 7)
• a(12)
• 12a

• y(x 2) 6(x 2)
• (x 2)(y 6)
• Here (x 2) was the GCF

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Step 2 Difference of Squares (count the number
of terms)

• Count the number of terms.
• If the polynomial has two terms (it is a
  binomial), then see if it is the difference of
  two squares.
• a2 b2 (a b)(a b)
• x2 9 (x 3)(x 3)
• Remember the sum of squares will not factor. a2
  b2

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Using FOIL we find the product of two binomials.
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Rewrite the polynomial as the product of a sum
and a difference.
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Conditions for Difference of Squares

• Must be a binomial with subtraction.
• First term must be a perfect square.
• (x)(x) x2
• Second term must be a perfect square (5)(5) 25

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Recognizing the Difference of Squares

• Must be a binomial with subtraction.
• First term must be a perfect square (p)(p) p2
• Second term must be a perfect square (10)(10)
  100

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Recognizing the Difference of Squares

• Must be a binomial with subtraction.
• First term must be a perfect square (3m)(3m)
  9m2
• Second term must be a perfect square (7)(7) 49

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Step 2 Difference of Squares

• 4x2 25
• 2x2 8
• b2 100
• y4 16

• (2x 5)(2x 5)
• 2(x2 4)
• Sum of Squares. Will not factor.
• (y2 4)(y2 4)
• (y2 4)(y 2)(y 2)

• 2(x 2)(x 2)

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Step 3 Trinomials

• Count the number of terms.
• If there are three terms
• Check to see if it is a perfect square trinomial.
• Grouping method. (long)
• Inspection (skill)

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Using FOIL we find the product of two binomials.
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Rewrite the perfect square trinomial as a
binomial squared.
So when you recognize this
you can write this.
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Recognizing a Perfect Square Trinomial

• First term must be a perfect square.
• (x)(x) x2
• Last term must be a perfect square.
• (5)(5) 25
• Middle term must be twice the product of the
  roots of the first and last term.
• (2)(5)(x) 10x

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Recognizing a Perfect Square Trinomial

• First term must be a perfect square.
• (m)(m) m2
• Last term must be a perfect square.
• (4)(4) 16
• Middle term must be twice the product of the
  roots of the first and last term.
• (2)(4)(m) 8m

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Recognizing a Perfect Square Trinomial

• First term must be a perfect square.
• (p)(p) p2
• Last term must be a perfect square.
• (9)(9) 81
• Middle term must be twice the product of the
  roots of the first and last term.
• (2)(-9)(p) -18p

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Recognizing a Perfect Square Trinomial
Not a perfect square trinomial.

• First term must be a perfect square.
• (6p)(6p) 36p2
• Last term must be a perfect square.
• (5)(5) 25
• Middle term must be twice the product of the
  roots of the first and last term.
• (2)(5)(6p) 60p ? 30p

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Using FOIL we find the product of two binomials.
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The two binomials represent the trinomial in
factored form.
Our job is to rewrite trinomials in factored form.
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Start with the trinomial and pretend that you
have a factorization.
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Factoring a Trinomial by Grouping
First list the factors of 24.
Rewrite with four terms.
Now add the factors.
25
1 24
2 12
14
3 8
11
10
4 6
Notice that 4 and 6 sum to the middle term.
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Factoring a Trinomial by Grouping
First list the factors of 24.
Rewrite with four terms.
Now add the factors.
25
1 24
2 12
14
3 8
11
10
4 6
Notice that 2 and 12 sum to the middle term.
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Step 3 Inspection

• Guess at the factorization until you get it
  right.
• Check with multiplication.

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Step 4 Grouping

• If the polynomial has more than three terms, try
  to factor by grouping.

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Step 5

• As a final check see if any of the factors you
  have written can be factored further.
• If you have overlooked a common factor you can
  catch it here.

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Getting Ready for Class

• What is the first step in factoring any
  polynomial?
• If a polynomial has four terms, what method of
  factoring should you try?
• If a polynomial has two terms, what method of
  factoring should you try?
• What is the last step in factoring a polynomial?

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Homework

• Problem Set 5.5
• Pages 327
• 1-75 odd

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Homework 5.5 5

• x2 6x 9
• Notice 9 is a perfect square
• (x 3)(x 3) (x 3)2
• Check with multiplication
• (x 3)(x 3) x2 3x 3x 9
• x2 6x 9

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Homework 5.5 15

• 9x2 12xy 4y2
• Notice 9 and 4 are perfect squares
• (3x 2)(3x 2) (3x 2)2
• Check with multiplication
• (3x 2)(3x 2) 9x2 - 6x - 6x 4
• 9x2 12xy 4y2

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Homework 5.5 25

• x4 16
• Difference of two Squares
• x4 16 (x2 4)(x2 4)
• (x2 4)(x 2)(x 2)

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Homework 5.5 35
First list the factors of 2(-38) -76.
Rewrite with four terms.
Now subtract the factors.
75
1 76
2 38
36
4 19
15
Notice that 4 and 19 do the job.
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Homework 5.5 45

• 49x2 9y2
• Sum of Squares
• Prime
• Will not factor

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Homework 5.5 55

• 3x2 35xy 82y2
• (3x y)(x y)
• (3x 41y)(x 2y)
• (3x 41y)(x 2y)

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Homework 5.5 65
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Homework 5.5 75

• 75. Write the number 57600 in scientific
  notation.
• Move the decimal point between the 5 and the 7.
• Count the number of places the decimal point was
  moved.

57600
57600
5.76104