§ 5.3 greatest common factors and factoring by grouping
TRANSCRIPT
§ 5.3
Greatest Common Factors and Factoring by Grouping
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 5.3
Factoring
bybxyaxa 4263 22
Factoring a polynomial means finding an equivalent expression that is a product. For example, when we take the polynomial
bayx 232 2
And write it as
we say that we have factored the polynomial. In factoring, we write asum as a product.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.3
Factoring
Factoring a Monomial from a Polynomial1) Determine the greatest common factor of all terms in the polynomial.
2) Express each term as the product of the GCF and its other factor.
3) Use the distributive property to factor out the GCF.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 5.3
Factoring
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
344453534 357049 zyxzyxzyx
. 7 333 zyx
. 357049 344453534 zyxzyxzyx Factor:
The GCF is
First, we determine the greatest common factor of the three terms.
Notice that the greatest integer that divides into 49, 70 and 35 (the coefficients of the terms) is 7. The variables raised to the smallest exponents are . and , 333 zyx
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 5.3
Factoring
xyzyxzyzyxxzzyx 5710777 33323332333
344453534 357049 zyxzyxzyx
Factor out the GCF
Express each term as the product of the GCF and its other factor
CONTINUECONTINUEDD
xyzyxzzyx 51077 22333
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.3
Factoring
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
xxx 10505 23
251055 2 xxxxx
. 10505 23 xxx Factor:
The GCF is 5x. Because the leading coefficient, -5, is negative, we factor out a common factor with a negative coefficient. We will factor out the negative of the GCF, or -5x.
Factor out the GCF 2105 2 xxx
Express each term as the product of the GCF and its other factor
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.3
Factoring by Grouping
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
yxyxx 7
. 7 yxyxx Factor:
Let’s identify the common binomial factor in each part of the problem.
The GCF, a binomial, is x + y.
The GCF, a binomial, is x + y.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.3
Factoring by Grouping
yxyxx 17
17 xyx
yxyxx 7
We factor out the common binomial factor as follows.
Factor out the GCF
This step, usually omitted, shows each term as the product of the GCF and its other factor, in that order.
CONTINUECONTINUEDD
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 5.3
Factoring by Grouping
Factoring by Grouping1) Group terms that have a common monomial factor. There will usually be two groups. Sometimes the terms must be rearranged.
2) Factor out the common monomial factor from each group.
3) Factor out the remaining common binomial factor (if one exists).
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.3
Factoring by Grouping
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
by .bxyaxa 4263 22
yaxa 22 63
Factor:
There is no factor other than 1 common to all terms. However, we can group terms that have a common factor:
Common factor is : Use -2b, rather than 2b, as the common factor: -2bx – 4by = -2b(x + 2y). In this way, the common binomial factor, x + 2y, appears.
bybx 42
yxayaxa 2363 222
+
23a
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 5.3
Factoring by Grouping
bybxyaxa 4263 22
bybxyaxa 4263 22
The voice balloons illustrate that it is sometimes necessary to use a factor with a negative coefficient to obtain a common binomial factor for the two groupings. We now factor the given polynomial as follows:
Group terms with common factors
bayx 232 2
yxbyxa 2223 2
CONTINUECONTINUEDD
Factor out the common factors from the grouped terms
Factor out the GCF
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 5.3
Factoring by Grouping
EXAMPLEEXAMPLE
Your local electronics store is having an end-of-the-year sale. The price on a large-screen television had been reduced by 30%. Now the sale price is reduced by another 30%. If x is the television’s original price, the sale price can be represented by
(x – 0.3x) – 0.3(x – 0.3x)
(a) Factor out (x – 0.3x) from each term. Then simplify the resulting expression.
(b) Use the simplified expression from part (a) to answer these questions. With a 30% reduction followed by a 30% reduction, is the television selling at 40% of its original price? If not, at what percentage of the original price is it selling?
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 5.3
Factoring by Grouping
(a) (x – 0.3x) – 0.3(x – 0.3x)
SOLUTIONSOLUTION
CONTINUECONTINUEDD
= 1(x – 0.3x) – 0.3(x – 0.3x)
= (x – 0.3x)(1 – 0.3)
= (x – 0.3x)(0.7)
= 0.7x – 0.21x
This step, usually omitted, shows each term as the product of the GCF and its other factor, in that order.
Factor out the GCF
Subtract
Distribute
= 0.49x Subtract
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 5.3
Factoring by Grouping
(b) With a 30% reduction, followed by another 30% reduction, the expression that represents the reduced price of the television simplifies to 0.49x. Therefore, this series of price reductions effectively gives a new price for the television at 49% its original price, not 40%.
CONTINUECONTINUEDD