Lesson 2.3 Read: Pages 131-134 Page 127: #1-61 (EOO)

Lesson 2.3

Read: Pages 131-134

Page 127: #1-61 (EOO)

Real Zeros of Polynomial Functions

Objective

Students will know how to use long division and synthetic division to divide polynomials by other polynomials, how to determine the number of zeros of a polynomial, and how to find real zeros of a polynomial function.

x 2 6x 3 19x 2 16x 4

Use long division to divide.

5x 2 17x 12Divide by

x 4

8x 4 5Divide by

2x 1

(1 3x 2 x 4 ) (3 2x x 2)

Use synthetic division to divide.

(2x 3 14x 2 20x 7) (x 6)

(5x 3 6x 8) (x 2)

(x 3 729) (x 9)

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to verify that the

expressions are equivalent. Verify the results algebraically.

y1 x 2 2x 1

x 3,

y2 x 12

x 3

Write the function in the form for the given value of k. Use a graphing utility to demonstrate that .

f (x) x 4 10x 2 2x 4,

f (x) (x k)q(x) r(x)

f (k) r

k 3

f (x) 3x 3 8x 2 10x 8,

k 2 2

Use the Remainder Theorem and synthetic division to evaluate the function at each given value. Use a graphing utility to verify your results.

f (x) 2x 3 7x 3

f (1)(a)

f ( 2)(b)

f1

2

(c)

f (2)(d)

Use synthetic division to show that x is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all the real zeros of the function.

48x 3 80x 2 41x 6 0,

x 2

3

(a) Verify the given factors of the function f, (b) find the remaining factors of f, (c) use your results to write the complete factorization of f, and (d) list all real zeros of f.

f (x) 8x 4 14x 3 71x 2 10x 24,

(x 2),(x 4)

Use the Rational Zero Test to list all possible rational zeros of f.

f (x) 4x 5 8x 4 5x 3 10x 2 x 2

Find all real zeros of the polynomial function.

f (x) x 4 x 3 29x 2 x 30

z(x) 4x 4 43x 2 9x 90

(a) Use the zero or root feature of the graphing utility to approximate (to the nearest thousandth) the zeros of the function, (b) determine one of the exact zeros and use synthetic division to verify your result, and (c) factor the polynomial completely.

f (s) s3 12s2 40s 24