6.3 Factoring Polynomials

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6.3 Factoring Polynomials
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• There are three common ways of factoring
• factoring by the greatest common monomial factor
• factoring following a pattern
• factoring by grouping or the box method

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Greatest Common Monomial Factor
GCF

• The greatest common factor or is the greatest
  integer that is a factor of each of the given
  integers.
• The greatest common monomial factor (GCMF) is the
  monomial with the numerical coefficient and
  the greatest that is a factor of each term
  of the polynomial.
• Always look for a greatest common monomial factor
  first, and then try the other two methods.

greatest
degree
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Factor out the greatest common monomial factor.

• 1.
• 2.

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Factoring by Finding a Pattern

•  Remember, always look for a GCMF before looking
  for a pattern.
• Perfect Square Trinomial Patterns
• For all a and b,
• and
• Difference of Squares Pattern (Note the sum of
  squares is factorable)
• For all a and b,

not
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Factor completely.

• 3.
• 4.

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Factoring by Finding a Pattern

• Sum of Cubes
• For all a and b,
• Difference of Cubes
• For all a and b,

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Factor Completely.

• 5.
• 6.

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Factoring by Grouping or the Box Method

• Factoring a polynomial in the form
• Example
• Factor completely .
• Draw a 2x2 box

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• Place the highest degree term in the top left box
  and the constant in the bottom right box.
• Now, multiply the two together, what do you get?

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• Find factors of that product that add to the
  middle term of the trinomial.
• If the product is positive, the factors are
  either both positive or both negative
• If the product is negative, one factor is
  positive and one factor is negative.

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• Place each factor in one of the remaining boxes.
• Then, factor out the GCMF from each column and
  each row. If the top most term or left most term
  is negative, factor out the negative. If there is
  no GCMF, factor out a 1. Write your results as
  two binomials multiplied in parentheses
• Use FOIL to check your answer!

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Factor completely.

• 7.
• 8.

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Your Turn!

• Factor 9-12

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• 9.
• 10.
• 11.
• 12.