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Polynomials: Algebraically finding exact zeros

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 =.

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Polynomials: Algebraically finding exact zeros

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  1. 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 = Looking at the graph and the list of possible rational zeros, suggests 4/3 might be a rational zero of P(x). ZOOM 4 Polynomials: Algebraically finding exact zeros Example 1: Find all exact zeros of the polynomial, P(x) = 3x3 – 16x2 + 19x – 4.

  2. The rational zero can be confirmed by pressing 2nd TRACE ENTER and entering 4/3, then . ENTER 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)(3x2– 12x + 3). Polynomials: Algebraically finding exact zeros Since 4/3 is a zero, (x – 4/3) is a factor, so next, divide P(x) by (x – 4/3). Slide 2

  3. The exact zeros are 4/3, and Note also P(x) factors as: P(x) = Polynomials: Algebraically finding exact zeros Remember, finding zeros of a polynomial, P(x) means solving the equation P(x) = 0. Therefore, to find the remaining two zeros, solve 3x2– 12x + 3 = 0, or x2– 4x + 1 = 0. Using the quadratic formula, Slide 3

  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 = Looking at the graph and the list of possible rational zeros, suggests - 1/2 and 1 are rational zeros of P(x). ZOOM 4 Polynomials: Algebraically finding exact zeros Example 2: Find all exact zeros of the polynomial, P(x) = 2x4 – x3 + 7x2 – 4x – 4. Slide 4

  5. The rational zeros can be confirmed by pressing and pressing the and keys. TRACE   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)(2x3 + x2 + 8x + 4). Polynomials: Algebraically finding exact zeros Since 1 is a zero, (x – 1) is a factor, so next, divide P(x) by (x – 1). Note,- 1/2 is also a zero, so P(x) can furthermore be written as: P(x) = (x– 1)(x + 1/2)(quadratic factor). Slide 5

  6. 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)(2x2+ 8). Polynomials: Algebraically finding exact zeros To find the unknown quadratic factor, divide 2x3+ x2+ 8x + 4 by x + 1/2. Last, find the remaining two zeros by solving 2x2+ 8 = 0, or x2+ 4 = 0. x=  2i x2+ 4 = 0, x2= - 4, The exact zeros are - 1/2, 1, - 2i, and 2i. Slide 6

  7. The exact zeros are -3, 1, and Polynomials: Algebraically finding exact zeros Try: Find all exact zeros of the polynomial, P(x) = 4x3 – 5x2 + 14x + 15. The exact zeros are - 3/4, 1 + 2i, and 1 – 2i. Try: Find all exact zeros of the polynomial, P(x) = x4 – 11x2 – 2x + 12. Slide 7

  8. Polynomials: Algebraically finding exact zeros END OF PRESENTATION Click to rerun the slideshow.

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