2.3 real zeros of polynomial functions 2014/15 digital lesson

2.3Real Zeros of

Polynomial Functions2014/15

Digital Lesson

HWQ

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

4 24f x x x

Find the zeros of the function andSketch a graph with those zeros and correct end

behavior.

Objective

• To find real zeros of polynomial functions using various methods.



A real number a is a zero of a function y = f (x)if and only if f (a) = 0.

A polynomial function of degree n has at most n zeros.

Real Zeros of Polynomial Functions

If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent.

1. x = a is a zero of f.

2. x = a is a solution of the polynomial equation f (x) = 0.

3. (x – a) is a factor of the polynomial f (x).

4. (a, 0) is an x-intercept of the graph of y = f (x).


+ 2 2 3 1 2 xxx

Example: Divide x2 + 3x – 2 by x + 1 and check the answer.

x

x2 + x2x – 22x + 2

– 4

remainder

Check: (x + 2)quotient

(x + 1)divisor

+ (– 4)remainder

= x2 + 3x – 2 dividend

Answer: x + 2 +1x

– 4

Dividing Polynomials

You try:

Divide by using long division.


22 10 12x x 3x

Try another one:Divide using long division.


( ) ( )x x x x4 2 23 1 2 3


16

Synthetic division is a shorter method of dividing polynomials. This method can be used only when the divisor is of the form x – a. It uses the coefficients of each term in the dividend.

Example: Divide 3x2 + 2x – 1 by x – 2 using synthetic division.

3 2 – 12

Since the divisor is x – 2, a = 2.

3

6

8 15coefficients of quotient remainder

value of a coefficients of the dividend

3x + 8Answer: 2

x15

1. Divide using synthetic division.


2 10 12 32x x x


Factor Theorem: A polynomial f(x) has a factor (x – k) if and only if f(k) = 0.

Example: Show that (x + 2) and (x – 1) are factors of f(x) = 2x

3 + x2 – 5x + 2.

62 1 – 5 2– 2

2

– 4

– 3 1

– 2

0

The remainders of 0 indicate that (x + 2) and (x – 1) are factors.

– 12 – 3 11

2

2

– 1 0

The complete factorization of f is (x + 2)(x – 1)(2x – 1).

Find the real zeros of the function.

• f(x) = 2x3 + x2 – 5x + 2.



Graphing Utility: Find the zeros of f(x) = 2x3 + x2 – 5x + 2.

Calc Menu:

The zeros of f(x) are x = – 2, x = 0.5, and x = 1.

– 10 10

10

– 10

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.

x=2


x x3 7 6 0

Use the graphing calculator to find the one exact zero. Then use synthetic division to break down the function to find the other two exact zeros.


f x x x x( ) 3 22 7 2

• Homework:• Pg. 123: 1-33 EOO (every other odd),

35-43 EOO, 49-53 odd


2.3Real Zeros of

Polynomial FunctionsDay 2 – More

problems

Digital Lesson

Verify that (x+2) and (x-4) are factors of the function and find any remaining factors to write the complete factorization of f. List all real zeros.


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

HWQ practice

Solve by dividing out a zero and factoring the resulting quadratic ( or use quadratic formula):


3 26 11 24 9 0x x x

Use the graphing calculator to find the any exact zeros. Then use synthetic division to break down the function to find the other zeros.


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

Find all real solutions of the polynomial. Use the rational root test, or graphing calculator. Extra credit to anyone who can factor the polynomial by hand.


5 4 3 23 5 2 0x x x x x

• Homework: 2.1/2.3 Review Worksheet


2.3 real zeros of polynomial functions 2014/15 digital lesson

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