Connection Suppose you know that the area of a certain rectangle is 3xy – 21y + 4x – 35 square feet. Is it possible for this rectangle to have dimensions that can be represented with integral coefficients? If so, what are these dimensions? Since the area of a rectangle is the product of its length and width, you can determine an answer to the questions above if you can find two binomials with integral coefficients whose product is 3xy – 21y + 5x – 35. You can do this by factoring.
Yeah, but then what? Polynomials with four or more terms like 3xy – 21y + 5x – 35, can sometimes be factored by grouping terms of the polynomials. One method is to group the terms into binomials that can be factored using the distributive property. Then use the distributive property again with a binomial as the common factor. Look!
Ex. 1: Factor 3xy – 21y + 5x – 35 to answer the questions presented above. Group terms that have common monomial factor. Factor. Notice that (x – 7) is a common factor. Distributive property. Check by using FOIL. Thus, the dimensions of this rectangle can be represented by binomials with integral coefficients. These dimensions would be 3y+5 feet and x – 7 feet.
Ex. 2: Factor Group terms that have common monomial factor. Factor. Notice that (8mn - 5) is a common factor. Distributive property. Check by using FOIL. Sometimes you can group the terms in more than one way when factoring a polynomial. For example, the polynomial in Example 2 could have been factored in a different manner.
Note: Recognizing binomials that are additive inverses is often helpful in factoring. For example, the binomials 3 – a and a – 3 are additive inverses since the sum of 3 – a and a – 3 is 0. Thus, 3 – a and –a +3 are equivalent. What is the additive inverse of 5 – y? -y + 5
Ex. 4: Factor: FOIL AND CHECK (5-y) and (y-5) are additive inverses. (5-y)=(-1)(y-5)