5.12: Solving Equations by Factoring Zero Product Property: For all real numbers a and b: ab = 0 if and only if a = 0 or b = 0. (Multiplicative Property of Zero) (A product of factors is zero if and only if one or more of the factors is zero). (Converse) -The zero-product property is true for any number of factors. You can use this property to solve certain equations. Polynomial Equations: both sides of an equation are polynomials. Polynomial equations are usually named by the term of highest degree. If a ≠0: + = 0 is a linear equation. 2 + + = 0 is a quadratic equation. 3 + 2 + + = 0 is a cubic equation. -Many polynomial equations can be solved by factoring and then using the zero-product property. Often, the first step is to transform the equation into standard form in which one side is zero. The other side should be a simplified polynomial arranged in order of decreasing degree of the variable. -If a factor occurs twice in the factored form of an equation, it is a double root or multiple root. It is only listed one time in the solution set. Ex: #3, p.232 ( + ) = 1. Set each factor equal to zero and solve. 15( + 15) = 0 ( + 15) = 0 15 = 0 =0 = −15 The solution set is {0, −15}. Ex: #15, p.232 = + 1. Transform the equation into standard form. 2. Factor the polynomial. 2 − 4 − 32 = 0 ( 2 − 8) + (4 − 32) = 0 ( − 8) + 4( − 8) = 0 ( − 8) ( + 4) = 0 −8 = 0 +4 =0 =8 = −4 The solution set is {8, −4}. 3. Set each factor equal to zero and solve. Ex: #39, p.232 + = 1. Transform the equation into standard form. 2. Factor completely. 3. Set each factor equal to zero and solve. 9 3 − 30 2 + 9 = 0 3 ( 3 2 − 10 + 3) = 0 3 ( 3 2 − 9 − + 3) = 0 3 {(3 2 − 9 + (− + 3)} = 0 3 {3( − 3) − 1( − 3)} = 0 3 (3 − 1) ( − 3) = 0 3 = 0 3 − 1 = 0 −3 = 0 =0 = 1 3 The solution set is {0, =3 1 , 3 3}. 5.12: Solving Equations by Factoring ( − ) ( + ) = Ex: #45, p.233 1. Since you have factors that are equal to something besides zero, you must turn this into a polynomial (by foiling). 2 + 3 − 2 − 6 = 6 2. Transform the equation into standard form. 2 + − 12 = 0 {( 2 + 4) + (−3 − 12)} = 0 3. Factor the polynomial. {( + 4) − 3( + 4)} = 0 ( + 4) ( − 3) = 0 4. Set each factor equal to zero and solve. +4=0 −3=0 = −4 =3 The solution set is {−4, 3}.

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# 2) 5.12: Solving Equations by Factoring