a. Which model better describes the data for 2007? b. Does the polynomial model of degree 2 underestimate or overestimate the average TV price for 2012? By how much? SECTION P.5 Factoring Polynomials Objectives 1 Factor out the greatest common factor of a polynomial.
8/24/2016 Algebra and Trigonometry 115/171 2 Factor by grouping. 3 Factor trinomials. 4 Factor the difference of squares. 5 Factor perfect square trinomials. 6 Factor the sum or difference of two cubes. 7 Use a general strategy for factoring polynomials. 8 Factor algebraic expressions containing fractional and negative exponents. A two-year-old boy is asked, “Do you have a brother?” He answers, “Yes.” “What is your brother’s name?” “Tom.” Asked if Tom has a brother, the two-year- old replies, “No.” The child can go in the direction from self to brother, but he cannot reverse this direction and move from brother back to self. As our intellects develop, we learn to reverse the direction of our thinking. Reversibility of thought is found throughout algebra. For example, we can multiply polynomials and show that 5 x (2 x + 3) = 10 x 2 + 15 x . We can also reverse this process and express the resulting polynomial as 10 x 2 + 15 x = 5 x (2 x + 3). Factoring a polynomial expressed as the sum of monomials means finding an equivalent expression that is a product. In this section, we will be factoring over the set of integers , meaning that the coefficients in the factors are integers. Polynomials that cannot be factored using integer coefficients are called irreducible over the integers , or prime .
8/24/2016 Algebra and Trigonometry 116/171 The goal in factoring a polynomial is to use one or more factoring techniques until each of the polynomial’s factors, except possibly for a monomial factor, is prime or irreducible. In this situation, the polynomial is said to be factored completely . We will now discuss basic techniques for factoring polynomials. 1 Factor out the greatest common factor of a polynomial. Common Factors In any factoring problem, the first step is to look for the greatest common factor . The greatest common factor , abbreviated GCF, is an expression of the highest degree that divides each term of the polynomial. The distributive property in the reverse direction ab + ac = a ( b + c ) can be used to factor out the greatest common factor. EXAMPLE 1 Factoring Out the Greatest Common Factor Factor: a. 18 x 3 + 27 x 2 b. x 2 ( x + 3) + 5( x + 3). SOLUTION a. First, determine the greatest common factor. The GCF of the two terms of the polynomial is 9 x 2 . 18 x 3 + 27 x 2 = 9 x 2 (2 x ) + 9 x 2 (3) Express each term as the product of the GCF and its other factor. = 9 x 2 (2 x + 3) Factor out the GCF. b. In this situation, the greatest common factor is the common binomial factor ( x + 3). We factor out this common factor as follows: x 2 ( x + 3) + 5 ( x + 3) = ( x + 3) ( x 2 + 5).
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