Polynomial Review

The Basics

100

What is a polynomial?

A monomial or a sum or difference of monomials.

200

What is the degree of the following polynomial:
3x²-4+8x⁴

The degree is 4.

300

If h(x) = x³ - 2x² + 3x - 1, find h(-2)

h(-2) = -23

400

Subtract the following polynomials: 

(34 + 8x³ - 9x²) 

- (3x³ + 10x² - 4x - 4)

5x³-19x²+4x+38

500

Simplify the expression so that there are no negative exponents: 

(2a³b^-4)²(-4a²b⁴)

-16a⁸ / b⁴

Dividing Polynomials

100

If the remainder after synthetic division is NOT zero, is (x - a) a factor of P(x)?

No!

If the remainder is NOT zero, (x - a) is NOT a factor.

200

If (x + 1/2) is our divisor, what do we put in the little box in order to perform synthetic division?

-1/2

300

Divide the following polynomials:

(x³ +6x² +10x + 4) 

/ (x+2)

x² + 4x + 2

400

Find the remainder when: 

(-y² + 2y³ + 25) / (y - 3)

70

500

Divide the following polynomials:

(15x³ + 3x⁴ + 3x² -12x +4) 

/ (x² +4x - 2)

3x² +3x - 3 + (6x-2)/(x² +4x - 2)

Factoring Polynomials

100

Factor the following polynomial: 

x² - 17x - 38

(x + 2) (x - 19)

200

Factor the following polynomial: 

4x³ - 8x² + 9x - 18

(x - 2) (4x² + 9)

300

Factor:

x³ - 125

(x - 5)(x² + 5x + 25)

400

Factor the following polynomial: 

4d⁴ + 108d

4d (d³ + 27) =

4d (d + 3) (d² - 3d + 9)

500

Factor:

x⁴ +x² - 20

(x² + 5)(x² - 4) =

(x² + 5)(x + 2)(x - 2)

Finding those Roots

100

What is the difference between a "zero" of a polynomial and a "root" of a polynomial?

There is no difference!

200

Find the zeros of 

f(x) = 2x² + 5x + 3

x = -1 and x = -3/2

300

Is (x + 2) a factor of 

P(x) = -2x³ + 7x² - 3x - 9 ?

(x + 2) is NOT a factor of P(x) = -2x³ + 7x² - 3x - 9, because the remainder after division is 41, NOT zero.

400

Write a possible equation of a polynomial with roots at x = -2, 3, and 1. Write your answer in standard form.

y = x³ - 2x² - 5x + 6

500

Find all the roots of 

P(x) = x⁴ - x³ + 7x² - 9x - 18

x = -1, 2, 3i, and -3i