Factor greatest common factor from a polynomial'

1
Section 5.3

• Factor greatest common factor from a polynomial.
• Factor by grouping.

2

• We have seen that we use multiplication to get
• 4x2(2x 1) 8x3 4x2

3

• We have seen that we use multiplication to get
• 4x2(2x 1) 8x3 4x2
• What if we start with 8x3 4x2 and want to get
• 4x2(2x 1)?

4

• We have seen that we use multiplication to get
• 4x2(2x 1) 8x3 4x2
• What if we start with 8x3 4x2 and want to get
• 4x2(2x 1)?
• Factoring a polynomial means finding an
  equivalent expression that is a product.

5
Factoring a Monomial from a Polynomial

• Determine the greatest common factor of all terms
  in the polynomial.

6
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.

7
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.
• Use the distributive property to factor out the
  GCF.

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Factoring a Monomial from a Polynomial

• 12x2 9x

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Factoring a Monomial from a Polynomial

• 12x2 9x has GCF 3x
• Now factor the GCD out of each term

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Factoring a Monomial from a Polynomial

• 12x2 9x has GCF 3x
• Now factor the GCD out of each term
• 3x(4x 3)

11
Factoring a Monomial from a Polynomial

• 9x4 18x3 6x2

12
Factoring a Monomial from a Polynomial

• 9x4 18x3 6x2 has GCF 3x2

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Factoring a Monomial from a Polynomial

• 9x4 18x3 6x2 has GCF 3x2
• Now factor the GCD out of each term
• 3x2(3x2 6x 2)

14
Factoring a Monomial from a Polynomial

• 49x4y3z5 - 70x3y5z4 35x4y4z3

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Factoring a Monomial from a Polynomial

• 49x4y3z5 - 70x3y5z4 35x4y4z3 has GCF 7x3y3z3

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Factoring a Monomial from a Polynomial

• 49x4y3z5 - 70x3y5z4 35x4y4z3 has GCF 7x3y3z3
• Now factor the GCD out of each term
• 7x3y3z3(7xz2 - 10y2z 5xy)

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Factoring a Monomial from a Polynomial

• 5(x 4) a(x 4)

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Factoring a Monomial from a Polynomial

• 5(x 4) a(x 4) has GCD (x 4)

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Factoring a Monomial from a Polynomial

• 5(x 4) a(x 4) has GCD (x 4)
• Now factor out (x 4)
• (x 4)(5 a)

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Factoring a Monomial from a Polynomial

• 7x(x y) (x y)

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Factoring a Monomial from a Polynomial

• 7x(x y) (x y) has GCD (x y)

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Factoring a Monomial from a Polynomial

• 7x(x y) (x y) has GCD (x y)
• Now factor out (x y)
• (x y)(7x 1)

23
Factoring a Monomial from a Polynomial

• (x4)(x 5) (x 1)(x 5)

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Factoring a Monomial from a Polynomial

• (x4)(x 5) (x 1)(x 5) has GCD (x 5)

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Factoring a Monomial from a Polynomial

• (x4)(x 5) (x 1)(x 5) has GCD (x 5)
• Now factor out (x 5)
• (x 5)((x 4) (x 1))

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Factoring a Monomial from a Polynomial

• (x4)(x 5) (x 1)(x 5) has GCD (x 5)
• Now factor out (x 5)
• (x 5)((x 4) (x 1))
• (x 5)(2x 3)

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Factor By Grouping

• Some polynomials have only 1 as a GCD. It may
  still be possible to factor by grouping terms
  with common divisors.

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Factor By Grouping

• x2 2x 4x 8

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Factor By Grouping

• x2 2x 4x 8 has GCD 1

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Factor By Grouping

• x2 2x 4x 8 has GCD 1
• x2 2x 4x 8 (x2 2x) (4x 8)

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Factor By Grouping

• x2 2x 4x 8 has GCD 1
• x2 2x 4x 8 (x2 2x) (4x 8)
• x(x 2) 4(x 2)

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Factor By Grouping

• x2 2x 4x 8 has GCD 1
• x2 2x 4x 8 (x2 2x) (4x 8)
• x(x 2) 4(x 2)
• Now factor out the GCD (x 2)

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Factor By Grouping

• x2 2x 4x 8 has GCD 1
• x2 2x 4x 8 (x2 2x) (4x 8)
• x(x 2) 4(x 2)
• Now factor out the GCD (x 2)
• (x 2)(x 4)

34
Factor By Grouping

• What if we had grouped differently?
• x2 2x 4x 8 has GCD 1
• x2 2x 4x 8 (x2 4x) (2x 8)

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Factor By Grouping

• What if we had grouped differently?
• x2 2x 4x 8 has GCD 1
• x2 2x 4x 8 (x2 4x) (2x 8)
• x(x 4) 2(x 4)

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Factor By Grouping

• What if we had grouped differently?
• x2 2x 4x 8 has GCD 1
• x2 2x 4x 8 (x2 4x) (2x 8)
• x(x 4) 2(x 4)
• Now factor out the GCD (x 4)

37
Factor By Grouping

• What if we had grouped differently?
• x2 2x 4x 8 has GCD 1
• x2 2x 4x 8 (x2 4x) (2x 8)
• x(x 4) 2(x 4)
• Now factor out the GCD (x 4)
• (x 4)(x 2)

38
Factor By Grouping

• Sometimes you must rearrange your terms.

39
Factor By Grouping

• Sometimes you must rearrange your terms.
• 3x2 8y 12x 2xy has GCD 1

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Factor By Grouping

• Sometimes you must rearrange your terms.
• 3x2 8y 12x 2xy has GCD 1
• Note the first 2 terms have GCD 1 and thus should
  not be grouped.

41
Factor By Grouping

• Sometimes you must rearrange your terms.
• 3x2 8y 12x 2xy has GCD 1
• Note the first 2 terms have GCD 1 and thus should
  not be grouped.
• 3x2 8y 12x 2xy (3x2 12x) (-2xy
  8y)

42
Factor By Grouping

• Sometimes you must rearrange your terms.
• 3x2 8y 12x 2xy has GCD 1
• Note the first 2 terms have GCD 1 and thus should
  not be grouped.
• 3x2 8y 12x 2xy (3x2 12x) (-2xy
  8y) 3x(x 4) 2y(x 4)

43
Factor By Grouping

• Sometimes you must rearrange your terms.
• 3x2 8y 12x 2xy has GCD 1
• Note the first 2 terms have GCD 1 and thus should
  not be grouped.
• 3x2 8y 12x 2xy (3x2 12x) (-2xy
  8y) 3x(x 4) 2y(x 4)
• Now factor out the GCD (x 4)

44
Factor By Grouping

• Sometimes you must rearrange your terms.
• 3x2 8y 12x 2xy has GCD 1
• Note the first 2 terms have GCD 1 and thus should
  not be grouped.
• 3x2 8y 12x 2xy (3x2 12x) (-2xy
  8y) 3x(x 4) 2y(x 4)
• Now factor out the GCD (x 4)
• (x 4)(3x 2y)

45
Factor By Grouping

• x3 2x2 5x 10

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Factor By Grouping

• x3 2x2 5x 10 has GCD 1

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Factor By Grouping

• x3 2x2 5x 10 has GCD 1
• x3 2x2 5x 10 (x3 2x2) (5x 10)

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Factor By Grouping

• x3 2x2 5x 10 has GCD 1
• x3 2x2 5x 10 (x3 2x2) (5x 10)
• x2(x 2) 5(x 2)

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Factor By Grouping

• x3 2x2 5x 10 has GCD 1
• x3 2x2 5x 10 (x3 2x2) (5x 10)
• x2(x 2) 5(x 2)
• Now factor out the GCD (x 2)

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Factor By Grouping

• x3 2x2 5x 10 has GCD 1
• x3 2x2 5x 10 (x3 2x2) (5x 10)
• x2(x 2) 5(x 2)
• Now factor out the GCD (x 2)
• (x 2)(x2 5)