1. Table of Contents Polynomials: Algebraically finding exact zeros Example 1:Find all exact zeros of the polynomial, P(x) = 3x 3 – 16x 2 + 19x – 4. First, use the Rational Zero Test to make up a list of possible rational zeros. (see separate slideshow) Second, graph the function in the decimal viewing window by pressing, entering the function, and pressing. Y = ZOOM 4 Looking at the graph and the list of possible rational zeros, suggests 4/3 might be a rational zero of P(x).

2. Table of Contents Polynomials: Algebraically finding exact zeros Slide 2 The rational zero can be confirmed by pressing 2ndTRACEENTER and entering 4/3, then. ENTER Since 4/3 is a zero, (x – 4/3) is a factor, so next, divide P(x) by (x – 4/3). 3 - 16 19 - 4 4 - 16 4 4/3 3 - 12 3 0 This means P(x) can be written as: P(x) = (x – 4/3)(3x 2 – 12x + 3).

3. Table of Contents Polynomials: Algebraically finding exact zeros Slide 3 Remember, finding zeros of a polynomial, P(x) means solving the equation P(x) = 0. Therefore, to find the remaining two zeros, solve 3x 2 – 12x + 3 = 0, or x 2 – 4x + 1 = 0. Using the quadratic formula, The exact zeros are 4/3, and Note also P(x) factors as: P(x) =

4. Table of Contents Polynomials: Algebraically finding exact zeros Slide 4 Example 2:Find all exact zeros of the polynomial, P(x) = 2x 4 – x 3 + 7x 2 – 4x – 4. First, use the Rational Zero Test to make up a list of possible rational zeros. (see separate slideshow) Second, graph the function in the decimal viewing window by pressing, entering the function, and pressing. Y = ZOOM 4 Looking at the graph and the list of possible rational zeros, suggests - 1/2 and 1 are rational zeros of P(x).

5. Table of Contents Polynomials: Algebraically finding exact zeros Slide 5 Since 1 is a zero, (x – 1) is a factor, so next, divide P(x) by (x – 1). 2 - 1 7 - 4 - 4 2 1 8 4 1 2 1 8 4 0 This means P(x) can be written as: P(x) = (x – 1)(2x 3 + x 2 + 8x + 4). The rational zeros can be confirmed by pressing and pressing the and keys. TRACE Note, - 1/2 is also a zero, so P(x) can furthermore be written as: P(x) = (x – 1)(x + 1/2)(quadratic factor).

6. Table of Contents Polynomials: Algebraically finding exact zeros Slide 6 To find the unknown quadratic factor, divide 2x 3 + x 2 + 8x + 4 by x + 1/2. 2 1 8 4 - 1 0 - 4 - 1/2 2 0 8 0 This means P(x) can be written as: P(x) = (x – 1)(x + 1/2)(2x 2 + 8). Last, find the remaining two zeros by solving 2x 2 + 8 = 0, or x 2 + 4 = 0. x 2 + 4 = 0,x 2 = - 4, x = 2i The exact zeros are - 1/2, 1, - 2i, and 2i.

7. Table of Contents Polynomials: Algebraically finding exact zeros Slide 7 Try:Find all exact zeros of the polynomial, P(x) = 4x 3 – 5x 2 + 14x + 15. The exact zeros are - 3/4,1 + 2i, and 1 – 2i. Try:Find all exact zeros of the polynomial, P(x) = x 4 – 11x 2 – 2x + 12. The exact zeros are -3, 1, and

8. Table of Contents Polynomials: Algebraically finding exact zeros