W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group

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Chabot Mathematics
5.3 GCFGrouping
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
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Review

• Any QUESTIONS About
• 5.2 ? PolyNomial Multiplication
• Any QUESTIONS About HomeWork
• 5.2 ? HW-17

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PolyNomial Factoring Defined

• To factor a polynomial is to find an equivalent
  expression that is a product. An equivalent
  expression of this type is called a factorization
  of the polynomial
• Factoring Breaks an algebraic expression into its
  simplest pieces
• Simplest ? Smallest Powers

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Example ? Factoring Monomials

• Find three factorizations of 24x3.
• SOLUTION
• a) 24x3 (6 ? 4)(x ? x2)
• 6x ? 4x2
• b) 24x3 (6 ? 4)(x2 ? x)
• 6x2 ? 4x
• c) 24x3 ((-6)(-4))x3
• (-6)(-4x3)

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Greatest Common Factor (GCF)

• Find the prime factorization of 105 60
• Use Factor-Tree

60
105
2
30
?
5
21
?
?
2
15
3
7
?
3
5
?
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Example ? GCF

• Thus

• Recognize the Factors that both numbers have in
  COMMON

• The GREATEST Common Factor is the PRODUCT of all
  the COMMON Factors
• In This Case the GCF

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Examples ? GCF

• Find the GCF for Monomials 14p4q and 35pq3
• The Prime Factorizations
• 14p4q 2 ? 7 ? p ? p ? p ? p ? q
• 35pq3 5 ? 7 ? p ? q ? q ? q

• Thus the GCF 7 ? p ? q 7pq

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Examples ? GCF

• Find the GCF for Three Monomials 15x2
  30xy2 57x3y
• The Prime Factorizations
• 15x2 3 ? 5 ? x ? x
• 30xy2 2 ? 3 ? 5 ? x ? y ? y
• 57x3y 3 ? 19 ? x ? x ? x ? y

ID the Common Factors

• Thus the GCF 3 ? x 3x

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Factoring When Terms Have a Common Factor

• To factor a polynomial with two or more terms of
  the form ab ac, we use the distributive law
  with the sides of the equation switched ab
  ac a(b c).
• Multiply Factor
• 4x(x2 3x - 4) 4x3 12x2 - 16x
• 4x?x2 4x?3x - 4x?4 4x?x2 4x?3x -
  4x?4
• 4x3 12x2 - 16x 4x(x2 3x - 4)

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Example ? Factor by Distributive

• Factor 9a - 21
• SOLUTION
• The prime factorization of 9a is 3?3?a
• The prime factorization of 21 is 3?7
• The largest common factor is 3.
• 9a - 21 3?3a - 3?7 (UNdist the 3)
• 3(3a - 7)
• Chk 3(3a - 7) 3 ? 3a - 3 ? 7 9a - 21

?
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Example ? Factor by Distributive

• Factor 28x6 32x3.
• SOLUTION
• The prime factorization of 28x6 is
• 2 ? 2 ? 7 ? x ? x ? x ? x ? x ? x
• The prime factorization of 32x3 is
• 2 ? 2 ? 2 ? 2 ? 2 ? x ? x ? x
• The largest common factor is 2 ? 2 ? x ? x ? x
  or 4x3.
• 28x6 32x3 (4x3 ? 7x ) (4x3 ? 8)
• 4x3(7x3 8)

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Factor 12x5 - 21x4 24x3

• The prime factorization of 12x5 is
• 2 ? 2 ? 3 ? x ? x ? x ? x ? x
• The prime factorization of 21x4 is
• 3 ? 7 ? x ? x ? x ? x
• The prime factorization of 24x3 is
• 2 ? 2 ? 2 ? 3 ? x ? x ? x
• The largest common factor is 3 ? x ? x ? x or
  3x3.
• 12x5 21x4 24x3 3x3 ? 4x2 3x3 ? 7x 3x3 ?
  8

3 ? x ? x ? x ? 2 ? 2 ? x ? x
3 ? x ? x ? x ? 7 ? x
3 ? x ? x ? x ? 2 ? 2 ? 2
3x3(4x2 7x 8)
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Example ? Distributive factoring

• Factor 9a3b4 18a2b3
• SOLUTION
• The Prime Factorizations

• The Greatest Common Factor is 9a2b3
• Distributing OUT the GCF Produces the
  factorization
• 9a3b4 18a2b3 9a2b3(ab 2)

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Example ? Distributive factoring

• Factor -4xy 8xw - 12x
• SOLUTION
• The Expanded Factorizations
• -4xy -4x ? y
• 8xw - 2 ? -4x ? w
• - 12x 3 ? -4x
• Thus the Factored expression
• -4xy 8xw - 12x -4x(y - 2w 3)

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Factoring Out a Negative GCF

• When the coefficient of the term of greatest
  degree is negative, it is sometimes preferable to
  factor out the -1 that is understood along with
  the GCF
• e.g. Factor Out the GCF for

Factor out only the 3 .
Both areCorrect
Or factor out the 3
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PolyNomial Factoring Tips

• Factor out the Greatest Common Factor (GCF), if
  one exists.
• The GCF multiplies a polynomial with the same
  number of terms as the original polynomial.
• Factoring can always be checked by multiplying.
• Multiplication should yield the original
  polynomial.

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Factoring by GROUPING

• Sometimes algebraic expressions contain a common
  factor with two or more terms.
• Example Factor x2(x 2) 3(x 2)
• SOLUTION The binomial (x 2) is a factor of
  BOTH x2(x 2) 3(x 2).
• Thus, (x 2) is a common factor so
• x2(x 2) 3(x 2) (x 2)x2 (x 2)3
• (x 2)(x2 3)

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Grouping Game Plan

• If a polynomial can be split into groups of terms
  and the groups share a common factor, then the
  original polynomial can be factored.
• This method, known as factoring by grouping, can
  be tried on any polynomial with four or more
  terms

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Examples ? Grouping

• Factor by grouping.
• a) 3x3 9x2 x 3
• b) 9x4 6x - 27x3 - 18
• Solution
• a) 3x3 9x2 x 3 (3x3 9x2) (x 3)
• 3x2(x 3) 1(x 3)
• (x 3)(3x2 1)

Dont Forget the 1
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Examples ? Grouping

• Factor by grouping.
• a) 3x3 9x2 x 3
• b) 9x4 6x - 27x3 - 18
• Solution
• b) 9x4 6x - 27x3 - 18
• (9x4 6x) (-27x3 - 18)
• 3x(3x3 2) (-9)(3x3 2)
• (3x3 2)(3x - 9)

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Example ? Grouping

• Factor y5 5y3 3y2 15
• SOLUTION

y5 5y3 3y2 15
(y5 5y 3) (3y 2 15)
Grouping
Factoring each binomial
y 3 (y 2 5) 3(y 2 5)
Factoring out the common factor(a BiNomial)
(y 2 5) (y 3 3)
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Factor 4ab 2ac 8xb 4xc

• Try grouping terms which have something in
  common. Often, this can be done in more than one
  way.
• For example

Grp-1
Grp-2
or
as xs Grouping
bs cs Grouping
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Factor 4ab 2ac 8xb 4xc

• Next, find the greatest common factor for the
  polynomial in each set of parentheses.

Grouping Set-1
Grouping Set-2

• The GCF for (4ab 2ac) is 2a
• The GCF for (8xb 4xc) is 4x

• The GCF for (4ab 8xb) is 4b
• The GCF for (2ac 4xc) is 2c

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Factor 4ab 2ac 8xb 4xc

• Write each of the polynomials in parentheses as
  the product of the GCF and the remaining
  polynomial

• Apply the distributive property to any common
  factors

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Factor 4ab 2ac 8xb 4xc

• Examine the Factorizations

• Notice that it did not matter how the terms were
  originally grouped, the factored forms of the
  polynomials are IDENTICAL

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WhiteBoard Work

• Problems From 5.3 Exercise Set
• 22, 32, 52, 56, 68, 84

• Factor byGrouping

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All Done for Today
Factoring4-TermPolynomials
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Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu

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Graph y x

• Make T-table

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Factor 4ab 2ac 8xb 4xc

• Divide each polynomial in parentheses by the GCF