2. Polynomial Equations Whenever two polynomials are set equal to each other, the result is a polynomial equation. In this section we learn how to solve polynomial equations both graphically and algebraically by factoring. Solving Polynomial Equations Graphically

3. Example Find the zeros of the function given by f(x) = x 3 – 2x 2 – 5x + 6. Solution Graph the equation, choosing a window that shows the x-intercepts of the graph. This may require several attempts. To find the zeros use the ZERO option from the CALC menu.

4. continued f(x) = x 3 – 2x 2 – 5x + 6 Use the same procedure for the other two zeros. Hence, the zeros/roots are x =  2, x = 1, and x = 3.

5. Solution Example

6. Solve: (x – 4)(x + 3) = 0. Solution According to the principle of zero products, at least one factor must be 0. x – 4 = 0 or x + 3 = 0 x = 4 or x =  3 For 4:For  3: (x – 4)(x + 3) = 0 (4 – 4)(4 + 3) = 0 (  3 – 4)(  3 + 3) = 0 0(7) = 0 0(  7) = 0 0 = 0 TRUE 0 = 0 TRUE (  3, 0) (4, 0) Solving Polynomial Equations Algebraically

8. Terms with Common Factors (GCF) When factoring a polynomial, we look for factors common to every term and then use the distributive law. MultiplyFactor 4x(x 2 + 3x  4) 4x 3 + 12x 2  16x = 4x  x 2 + 4x  3x  4x  4= 4x  x 2 + 4x  3x  4x  4 = 4x 3 + 12x 2  16x= 4x(x 2 + 3x  4) Example Factor: 28x 6 + 32x 3. Solution The prime factorization of 28x 6 is 2  2  7  x  x  x  x  x  x The prime factorization of 32x 3 is 2  2  2  2  2  x  x  x The greatest common factor is 2  2  x  x  x or 4x 3. 28x 6 + 32x 3 = 4x 3  7x 3 + 4x 3  8 = 4x 3 (7x 3 + 8)

9. Example Factor: 12x 5  21x 4 + 24x 3 Solution The prime factorization of 12x 5 is 2  2  3  x  x  x  x  x The prime factorization of 21x 4 is 3  7  x  x  x  x The prime factorization of 24x 3 is 2  2  2  3  x  x  x The greatest common factor is 3  x  x  x or 3x 3. 12x 5  21x 4 + 24x 3 = 3x 3  4x 2  3x 3  7x + 3x 3  8 = 3x 3 (4x 2  7x + 8)

10. Example Solution

11. Example Write an equivalent expression by factoring. a) 3x 3 + 9x 2 + x + 3 b) 9x 4 + 6x  27x 3  18 Solution a) 3x 3 + 9x 2 + x + 3 = (3x 3 + 9x 2 ) + (x + 3) = 3x 2 (x + 3) + 1(x + 3) = (x + 3)(3x 2 + 1) Don’t forget to include the 1. Factoring by Grouping b) 9x 4 + 6x  27x 3  18 = (9x 4 + 6x) + (  27x 3  18) = 3x(3x 3 + 2) + (  9)(3x 3 + 2) = (3x 3 + 2)(3x  9) = (3x 3 + 2)3(x  3) = 3(3x 3 + 2)(x  3)

13. Factoring and Equations Example Solve: 7x 2 = 35x. Solution Use the principle of zero products if there is a 0 on one side of the equation and the other side is in factored form. 7x 2 = 35x 7x 2 – 35x = 0 Subtracting 35x. One side is now 0. 7x(x – 5) = 0Factoring x = 0 or x – 5 = 0 Use the principle of zero products x = 0 or x = 5 To Use the Principle of Zero Products 1. Write an equivalent equation with 0 on one side, using the addition principle. 2. Factor the polynomial completely. 3. Set each factor that is not a constant equal to 0. 4. Solve the resulting equations.

14. To Use the Principle of Zero Products 1. Write an equivalent equation with 0 on one side, using the addition principle. 2. Factor the polynomial completely. 3. Set each factor that is not a constant equal to 0. 4. Solve the resulting equations.