Section 2.4: Real Zeros of Polynomial Functions

download report

Transcript Section 2.4: Real Zeros of Polynomial Functions

April 7, 2015

Polynomial Long Division

Divide by w – 3. 3

x

3  2

x

2  4

x

 3

x

 3

x

 3

Synthetic Division

3

x

3  6

x

 2

Remainder Theorem

If dividing f(x) by x – a, f(a) will determine the remainder.

x

 7

x

 6

x – 4?

Use the Remainder Theorem to evaluate f (x) = 6x 3 5x 2 + 4x – 17 at x = 3.

Factor Theorem

When dividing polynomials, if the remainder is zero, then the divisor is a factor.

Use the Factor Theorem to determine whether x – 1 is a factor of f (x) = 2x 4 + 3x 2 – 5x + 7.

Using the Factor Theorem, verify that x + 4 is a factor of f (x) = 5x 4 + 16x 3 – 15x 2 + 8x + 16.

Finding Exact Irrational Zeros

Find the exact zeros for the function. Identify each zero as rational or irrational.

Writing Functions w/ Given Conditions

If the zeros of the function are -3, 7, and -1, and the leading coefficient is 4, write an equation for the function.

Rational Zeros Theorem

Given a function with constant of p and leading coefficient of q, all possible rational zeros can be found by 

factors of p factors of q

 2

x

3  7

x

2  8

x

 6 Find the possible rational zeros of  3

x

3  5

x

2  9

x

 16

Upper and Lower Bounds Test

When dividing polynomials, if the quotient polynomial has all non-negative coefficients, then the “k” value is an upper bound. If the quotient polynomial has alternating sign coefficients, then the “k” value is a lower bound.

Show that all real roots of the equation

x

3  10

x

2  12

x

 20 lie between - 4 and 4.