4 reversing the signs in the binomial factors

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4. Reversing the signs in the binomial factors reverses the sign of the middle term. bx , b . c a . b ax 2 + bx + c : 60 Chapter P Prerequisites: Fundamental Concepts of Algebra Check Point 5 Factor: Factoring a Trinomial in Two Variables Factor: Solution Step 1 Find two First terms whose product is Step 2 Find two Last terms whose product is The possible factorizations are and Step 3 Try various combinations of these factors. The correct factorization of is the one in which the sum of the O utside and I nside products is equal to Here is a list of possible factorizations: Thus, Use FOIL multiplication to check either of these factorizations. Check Point 6 Factor: Factoring the Difference of Two Squares A method for factoring the difference of two squares is obtained by reversing the special product for the sum and difference of two terms. 3 x 2 - 13 xy + 4 y 2 . 2 x 2 - 7 xy + 3 y 2 = 1 2 x - y 21 x - 3 y 2 or 1 x - 3 y 21 2 x - y 2 . Possible Factorizations of 2 x 2 7 xy 3 y 2 (2x+3y)(x+y) (2x+y)(x+3y) (2x-3y)(x-y) (2x-y)(x-3y) 2xy+3xy=5xy 6xy+xy=7xy –2xy-3xy=–5xy –6xy-xy=–7xy Sum of Outside and Inside Products (Should Equal 7 x y) This is the required middle term. - 7 xy . 2 x 2 - 7 xy + 3 y 2 1 - y 21 - 3 y 2 . 1 y 21 3 y 2 3 y 2 . 2 x 2 - 7 xy + 3 y 2 = 1 2 x 21 x 2 2 x 2 . 2 x 2 - 7 xy + 3 y 2 . EXAMPLE 6 6 x 2 + 19 x - 7. Factor the difference of squares. The Difference of Two Squares If and are real numbers, variables, or algebraic expressions, then In words: The difference of the squares of two terms factors as the product of a sum and a difference of those terms. A 2 - B 2 = 1 A + B 21 A - B 2 . B A Factoring the Difference of Two Squares Factor: a. b. 81 x 2 - 49. x 2 - 4 EXAMPLE 7
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Study Tip Factoring as is not a complete factorization. The second factor, is itself a difference of two squares and can be factored. x 2 - 9, 1 x 2 + 9 21 x 2 - 9 2 x 4 - 81 Section P.5 Factoring Polynomials 61 Solution We must express each term as the square of some monomial. Then we use the formula for factoring Check Point 7 Factor: a. b. We have seen that a polynomial is factored completely when it is written as the product of prime polynomials. To be sure that you have factored completely, check to see whether any factors with more than one term in the factored polynomial can be factored further. If so, continue factoring. A Repeated Factorization Factor completely: Solution Express as the difference of two squares. The factors are the sum and the difference of the expressions being squared. The factor is the difference of two squares and can be factored. The factors of are the sum and the difference of the expressions being squared. Check Point 8 Factor completely: Factoring Perfect Square Trinomials Our next factoring technique is obtained by reversing the special products for squaring binomials. The trinomials that are factored using this technique are called perfect square trinomials .
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