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P.5 56 Chapter P Prerequisites: Fundamental Concepts of Algebra Factoring Polynomials 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 We can also reverse this process and express the resulting polynomial as Factoring a polynomial containing 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 . 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. 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 can be used to factor out the greatest common factor. ab + ac = a 1 b + c 2 Sum of monomials Equivalent expression that is a product The factors of 10 x 2 + 15 x are 5 x and 2 x + 3. Factoring 10 x 2 15 x 10x 2 +15x=5x(2x+3) 10 x 2 + 15 x = 5 x 1 2 x + 3 2 . 5 x 1 2 x + 3 2 = 10 x 2 + 15 x . Objectives Factor out the greatest common factor of a polynomial. Factor by grouping. Factor trinomials. Factor the difference of squares. Factor perfect square trinomials. Factor the sum or difference of two cubes. Use a general strategy for factoring polynomials. Factor algebraic expressions containing fractional and negative exponents. S e c t i o n Factor out the greatest common factor of a polynomial.
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Discovery In Example 2, group the terms as follows: Factor out the greatest common factor from each group and complete the factoring process. Describe what happens.What can you conclude? 1 x 3 + 3 x 2 + 1 4 x 2 + 12 2 . Study Tip The variable part of the greatest common factor always contains the smallest power of a variable or algebraic expression that appears in all terms of the polynomial. Section P.5 Factoring Polynomials 57 Factoring Out the Greatest Common Factor Factor: a. b.
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