§ 5.3 Greatest Common Factors and Factoring by Grouping

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 5.3 Factoring Factoring a polynomial means finding an equivalent expression that is a product. For example, when we take the polynomial And write it as we say that we have factored the polynomial. In factoring, we write a sum as a product.

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 5.3 Factoring Factoring a Monomial from a Polynomial 1) 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 FactoringEXAMPLE SOLUTION 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 P 332

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 5.3 Factoring Factor out the GCF Express each term as the product of the GCF and its other factor CONTINUED P 332

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 5.6 A Strategy for Factoring Polynomials, page 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 and Difference of two cubes: (b) If there are three terms, If is the trinomial a perfect square trinomial use one of the adjacent forms: If the trinomial is not a perfect square trinomial, If a is equal to 1, use the trial and error If a is > than 1, use the grouping method (c) If there are four or more terms, try factoring by grouping. ///////////////////////////////////////////

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 5.3 FactoringEXAMPLE SOLUTION 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 Express each term as the product of the GCF and its other factor P 333 Important to factor the negative sign

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 5.2 Factoring Check Point 1 Factor Check Point 2a Factor P

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 5.2 Factoring Check Point 2b Factor Check Point 2c Factor P

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 5.2 Factoring Check Point 3 Factor P 334 Want a to be positive

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 5.3 Factoring by GroupingEXAMPLE SOLUTION Factor: Let’s identify the common binomial factor in each part of the problem. The GCF, a binomial, is x + y. P 334

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 5.3 Factoring by Grouping 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. CONTINUED

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 5.2 Factoring Check Point 4a Factor Check Point 4b Factor P 334

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 5.3 Factoring by Grouping 1) 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). P 334

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 5.3 Factoring by GroupingEXAMPLE SOLUTION 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. + P 334

Blitzer, Intermediate Algebra, 5e – Slide #16 Section 5.3 Factoring by Grouping 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 CONTINUED Factor out the common factors from the grouped terms Factor out the GCF P 334

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 5.2 Factoring Check Point 5 Factor Check Point 6 Factor P 335-6

DONE

Blitzer, Intermediate Algebra, 5e – Slide #19 Section 5.3 Factoring by GroupingEXAMPLE 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 #20 Section 5.3 Factoring by Grouping (a) (x – 0.3x) – 0.3(x – 0.3x) SOLUTION CONTINUED = 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 = 0.49xSubtract

Blitzer, Intermediate Algebra, 5e – Slide #21 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%. CONTINUED