6.3 factoring polynomials

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( )