#### 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.