§ 5.6 a general factoring strategy. blitzer, intermediate algebra, 4e – slide #81 a strategy for...
TRANSCRIPT
§ 5.6
A General Factoring Strategy
Blitzer, Intermediate Algebra, 4e – Slide #81
A Strategy for Factoring Polynomials
A Strategy for Factoring a Polynomial1) If there is a common factor, factor out the GCF or factor out a common factor with a negative coefficient.
2) Determine the number of terms in the polynomial and try factoring as follows:
(a) If there are two terms, can the binomial be factored by using one of the
following special forms?
Difference of two squares:
Sum of two cubes:
Difference of two cubes:
BABABA 22
2233 BABABABA
2233 BABABABA
Blitzer, Intermediate Algebra, 4e – Slide #82
A Strategy for Factoring Polynomials
A Strategy for Factoring a Polynomial (b) If there are three terms, is the trinomial a perfect square trinomial? If so,
factor by using one of the following forms:
If the trinomial is not a perfect square trinomial, try factoring by trial and error or grouping.
(c) If there are four or more terms, try factoring by grouping.
3) Check to see if any factors with more than one term in the factored polynomial can be factored further. If so, factor completely.
Remember to check the factored form by multiplying or by using the TABLE or GRAPH feature of a graphing utility.
CONTINUECONTINUEDD
222 2 BABABA
222 2 BABABA
Blitzer, Intermediate Algebra, 4e – Slide #83
Factoring a Polynomial
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
.x 324 3 Factor:
1) If there is a common factor, factor out the GCF. Because 3 is common to both terms, we factor it out.
Factor out the GCF 183324 33 xx
2) Determine the number of terms and factor accordingly. The factor has two terms. This binomial can be expressed as , so it can be factored as the difference of two cubes.
18 3 x 33 12 x
Blitzer, Intermediate Algebra, 4e – Slide #84
Factoring a Polynomial
3) Check to see if factors can be factored further. In this case, they cannot, so we have factored completely.
Rewrite as the difference of two cubes
Simplify
33 123 x
183324 33 xx
CONTINUECONTINUEDD
22 1122123 xxx
124123 2 xxx
Factor
Blitzer, Intermediate Algebra, 4e – Slide #85
Factoring a Polynomial
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
.yyx 54 232 Factor:
1) If there is a common factor, factor out the GCF. Because 2y is common to both terms, we factor it out.
Factor out the GCF 4454 162232 yxyyyx
2) Determine the number of terms and factor accordingly. The factor has two terms. This binomial can be expressed as , so it can be factored as the difference of two squares.
4416 yx 22224 yx
Blitzer, Intermediate Algebra, 4e – Slide #86
Factoring a Polynomial
3) Check to see if factors can be factored further. We note that is the difference of two squares, , so we continue factoring.
Rewrite as the difference of two squares
CONTINUECONTINUEDD
Factor
4454 162232 yxyyyx
222242 yxy
2222 442 yxyxy
224 yx 222 yx
Blitzer, Intermediate Algebra, 4e – Slide #87
Factoring a Polynomial
The previous factorization
CONTINUECONTINUEDD
Rewrite last factor as the difference of two squares
2222 442 yxyxy
54 232 yyx
2222 242 yxyxy
yxyxyxy 2242 22 Factor
Blitzer, Intermediate Algebra, 4e – Slide #88
Factoring a Polynomial
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
.yxyx 22 550125 Factor:
1) If there is a common factor, factor out the GCF. Because 5 is common to both terms, we factor it out.
Factor out the GCF
2) Determine the number of terms and factor accordingly. The factor has three terms and is a perfect square trinomial. We factor using .
2222 10255550125 yxyxyxyx
22 1025 yxyx 222 2 BABABA
Blitzer, Intermediate Algebra, 4e – Slide #89
Factoring a Polynomial
Factor out the GCF
22 550125 yxyx
22 5255 yyxx 222 2 BABABA
3) Check to see if factors can be factored further. In this case, they cannot, so we have factored completely.
CONTINUECONTINUEDD
22 10255 yxyx
255 yx
Rewrite the part in parentheses in
Factor
form