Welcome: Pass out Zeros of Higher Polynomials WKS

HW Key: p. 212: 60, 62 (even)

Intermediate Value Theorem

Objectives & HW: The students will be able to apply the Intermediate Value Theorem in determining if there is a zero between two given values of x and in approximating that zero. HW: Intermediate Value Theorem Problems (back of notesheet)

The Intermediate Value Theorem 3.6 Topics on the Theory of Polynomial Functions (1) The Intermediate Value Theorem The Intermediate Value Theorem for Polynomials Let P(x) be a polynomial function with real coefficients. If P (a) and P (b) have opposite signs, then there is at least one value of c between a and b for which P (c) = 0. Equivalently, the equation P (x) = 0 has at least one real root between a and b.

EXAMPLE: Approximating a Real Zero 3.6 Topics on the Theory of Polynomial Functions (1) EXAMPLE: Approximating a Real Zero a. Show that the polynomial function f (x) = x3 - 2x - 5 has a real zero between 2 and 3. b. Use the Intermediate Value Theorem to find an approximation for this real zero to the nearest tenth. a. Let us evaluate f (x) at 2 and 3. If f (2) and f (3) have opposite signs, then there is a real zero between 2 and 3. Using f (x) = x3 - 2x - 5, we obtain Solution This sign change shows that the polynomial function has a real zero between 2 and 3. and f (3) = 33 - 2 * 3 - 5 = 27 - 6 - 5 = 16. f (3) is positive. f (2) = 23 - 2 * 2 - 5 = 8 - 4 - 5 = -1 f (2) is negative. more

EXAMPLE: Approximating a Real Zero (cont.) 3.6 Topics on the Theory of Polynomial Functions (1) EXAMPLE: Approximating a Real Zero (cont.) a. Show that the polynomial function f (x) = x3 - 2x - 5 has a real zero between 2 and 3. b. Use the Intermediate Value Theorem to find an approximation for this real zero to the nearest tenth. Solution b. A numerical approach is to evaluate f at successive tenths between 2 and 3, looking for a sign change. This sign change will place the real zero between a pair of successive tenths. Use the table function (2nd Window) of your calculator and set TblStart to 2 and ΔTbl to .1. Look at the table values (2nd Graph) and scroll down until you see a sign change. X Y1 2 -1 2.1 0.061 Sign change Sign change The sign change indicates that f has a real zero between 2 and 2.1. more

EXAMPLE: Approximating a Real Zero (cont.) 3.6 Topics on the Theory of Polynomial Functions (1) EXAMPLE: Approximating a Real Zero (cont.) a. Show that the polynomial function f (x) = x3 - 2x - 5 has a real zero between 2 and 3. b. Use the Intermediate Value Theorem to find an approximation for this real zero to the nearest tenth. Solution b. We now follow a similar procedure to locate the real zero between successive hundredths. Use the table function (2nd Window) of your calculator and set TblStart to 2 and ΔTbl to .01. Look at the table values (2nd Graph) and scroll down until you see a sign change. f (2.00) = -1 f (2.04) = -0.590336 f (2.08) = -0.161088 f (2.01) = -0.899399 f (2.05) = -0.484875 f (2.09) = -0.050671 f (2.02) = -0.797592 f (2.06) = -0.378184 f (2.1) = 0.061 f (2.03) = -0.694573 f (2.07) = -0.270257 Sign change The sign change indicates that f has a real zero between 2.09 and 2.1. Correct to the nearest tenth, the zero is 2.1.

You Try!!! Approximating a Real Zero 3.6 Topics on the Theory of Polynomial Functions (1) You Try!!! Approximating a Real Zero a. Show that the polynomial function f (x) = 3x2 - 2x - 6 has a real zero between 1 and 2. b. Use the Intermediate Value Theorem to find an approximation for this real zero to the nearest tenth. a. Let us evaluate f (x) at 1 and 2. If f (1) and f (2) have opposite signs, then there is a real zero between 1 and 2. Using f (x) = 3x3 - 2x - 5, we obtain Solution This sign change shows that the polynomial function has a real zero between 1 and 2. and f (2) = 3*22 - 2 * 2 - 6 = 12 - 4 - 6 = 2. f (2) is positive. f (1) = 3*12 - 2 * 1 - 6 = 3 - 2 - 6 = -5 f (1) is negative. more

EXAMPLE: Approximating a Real Zero (cont.) 3.6 Topics on the Theory of Polynomial Functions (1) EXAMPLE: Approximating a Real Zero (cont.) a. Show that the polynomial function f (x) = 3x2 - 2x - 6 has a real zero between 1 and 2. b. Use the Intermediate Value Theorem to find an approximation for this real zero to the nearest tenth. Solution b. A numerical approach is to evaluate f at successive tenths between 1 and 2, looking for a sign change. This sign change will place the real zero between a pair of successive tenths. Use the table function (2nd Window) of your calculator and set TblStart to 1 and ΔTbl to .1. Look at the table values (2nd Graph) and scroll down until you see a sign change. X Y1 1.7 -0.73 1.8 0.12 Sign change Sign change The sign change indicates that f has a real zero between 1.7 and 1.8. more

EXAMPLE: Approximating a Real Zero (cont.) 3.6 Topics on the Theory of Polynomial Functions (1) EXAMPLE: Approximating a Real Zero (cont.) a. Show that the polynomial function f (x) = 3x2 - 2x - 6 has a real zero between 1 and 2. b. Use the Intermediate Value Theorem to find an approximation for this real zero to the nearest tenth. Solution b. We now follow a similar procedure to locate the real zero between successive hundredths. Use the table function (2nd Window) of your calculator and set TblStart to 1 and ΔTbl to .01. Look at the table values (2nd Graph) and scroll down until you see a sign change. f (1.70) = -0.73 f (1.74) = -0.3972 f (1.78) = -0.0548 f (1.71) = -0.6477 f (1.75) = -0.3125 f (1.79) = 0.0323 f (1.72) = -0.5648 f (1.76) = -0.2272 f (1.80) = 0.12 f (1.73) = -0.4813 f (1.77) = -0.14.13 Sign change The sign change indicates that f has a real zero between 1.78 and 1.79. Correct to the nearest tenth, the zero is 1.8.

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