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6.3 Factoring Polynomials. 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. Greatest Common Monomial Factor. GCF. - PowerPoint PPT Presentation
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There are three common ways of factoring:factoring by the greatest common monomial
factorfactoring following a patternfactoring by grouping or the box method
Greatest Common Monomial FactorThe 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.
GCF
greatestdegree
Factor out the greatest common monomial factor.
1.
2.
6x5 −18x3 + 42x2
−62x5y2 +124x4y3
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,
a2 + 2ab+b2 = a+b( )2
a2 −2ab+b2 = a−b( )2
not
a2 −b2 = a+b( ) a−b( )
Factor completely. 3.
4.
x2 +12x+ 36
9m2n2 −49
Factoring by Finding a PatternSum of Cubes
For all a and b,
Difference of CubesFor all a and b,
a3 +b3 = a+b( ) a2 −ab+b2( )
a3 −b3 = a−b( ) a2 + ab+b2( )
Factor Completely.5.
6.
64 + 27c3
8n4c−17nc4
Factoring by Grouping or the Box Method Factoring a polynomial in the form
Example:Factor completely .
Draw a 2x2 box:
ax2 +bx+ c
6y2 −7y−5
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?
6y2 −7y−5
6y2
−5
−30y2
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.
−30y2
−10y +3y=−7y
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!
6y2
−5−10y3y
2y 13y
−5
2y +1( ) 3y−5( )
Factor completely. 7.
8.
2x2 + 5x−3
2x2 −2x−112
Your Turn!
Factor #9-12
9.
10.
11.
12.
3x−2( ) 3x−1( )
3x + 2y( ) 2x−3y( )
3g g−5h( )2
x2 −9( ) x2 −1( )
x+ 3( ) x−3( ) x+1( ) x−1( )