Lesson 2.3 Real Zeros of Polynomials

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# Lesson 2.3 Real Zeros of Polynomials - PowerPoint PPT Presentation

Lesson 2.3 Real Zeros of Polynomials. The Division Algorithm. Dividing by a polynomial Set up in long division. 2 terms in divisor (x + 1). How does this go into 1 st two terms in order to eliminate the 1 st term of the dividend. 2x. + 1. Multiply by the divisor

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### Lesson 2.3Real Zeros of Polynomials

Dividing by a polynomial

Set up in long division

2 terms in divisor (x + 1).

How does this go into 1st

two terms in order to

eliminate the 1st term of

the dividend.

2x

+ 1

• Multiply by the divisor
• Write product under dividend
• Subtract
• Carry down next term
• Repeat process

-

2x2 + 2x

-

x + 5

-

x + 1

-

4

HINTS:

If a term is missing in the dividend – add a “0” term.

If there is a remainder, put it over the divisor and add it to the quotient (answer)

Example 1

(x4 – x2 + x) ÷ (x2 - x + 1)

Synthetic Division

• Less writing
• Setting Up
• Divisor must be of the form: x – a
• Use only “a” and coefficients of dividend
• Write in “zero terms”

x – 2: a = 2

x + 3: a = -3

4 5 0 -2 5

Steps

• Bring down
• Multiply diagonally
• Numbers at bottom are coefficients

REPEAT

The Remainder Theorem

If f(x) is divided by x – a , the remainder is

r = f(a)

The Factor Theorem

If f(x) has a factor (x – a) then f(a) = 0

Rational Zero Test

Every rational zero =

Factors of constant term

=

Descartes’ Rule

Number of positivereal roots is:

► the number of variations in the signs, or

► less than that by a positive even integer

5x4 – 3x3 + 2x2 – 7x + 1

variations:

possible positive real roots:

Example 5

List possible zeros, verify with your calculator which are zeros, and check results with Descartes’ Rule

Problems Set 2.3