Polynomials & Polynomial Equations

Classifying Polynomials

100

Rewrite in standard form: 5x² + 4 - 2x³ - 4x²

-2x³ + x² + 4

200

Classify by degree and number of terms: -9x²

Quadratic monomial

300

Classify by degree and number of terms: 4x⁴ - 5x + 1

Quartic trinomial

400

Describe the end behavior of the polynomial function: y = 3x⁷ + 2x⁴ + 3x - 1

Down, up

500

Describe the end behavior and state the maximum number of turning points the graph will have: y = 3x¹² + 2x⁶ - x + 1

Up, up; 11

Factoring Polynomials

100

Find the zeros and state the multiplicity of multiple zeros: y = (x - 3)(x + 1)²(x - 6)

{-1 (multiplicity 2), 3, 6}

200

Find the zeros: y= -x² + 7x + 18

{-2, 9}

300

Write an equation in standard form having the given roots: -4, 6

y = x² - 2x - 24

400

Find the zeros: x³ - 6x² - 7x = 0

{-1, 0, 7}

500

Factor completely: 8x³ + 27

(2x + 3)(4x² - 6x + 9)

Polynomial Division

100

Find the quotient: (3x³ + 9x² + 8x + 4) ÷ (x + 2)

3x² + 3x + 2

200

Find the quotient: (6x² + 7x + 2) ÷ (2x + 1)

3x + 2

300

Find the quotient: (5x⁴ + x³+ 6x + 2) ÷ (x + 1)

5x³ - 4x² + 4x + 2

400

Find P(-2): P(x) = x³ + 7x² + 4x

12

500

Write the quotient and remainder as a single expression: (15x³ + x² - 21x) ÷ (5x - 3)

3x³ + 2x - 3 - 9/(5x - 3)

Rational Root Theorem

100

List the possible rational roots: x³ + 2x - 9 = 0

(all ±) 1, 3, 9

200

List the possible rational roots: 2x³ - 5x + 4 = 0

(all ±) 1, 2, 4, 1/2

300

List the possible rational roots: 8x³ + 2x² - 5x + 2 = 0

(all ±) 1, 2, 1/2, 1/4, 1/8

400

List the possible rational roots: 3x⁴ - 5x² - 2x - 6 = 0

(all ±) 1, 2, 3, 6, 1/3, 2/3

500

List the possible rational roots: 10x³ - 7x² + x - 10 = 0

(all ±) 1, 2, 5, 10, 1/2, 5/2, 1/5, 2/5, 1/10

Conjugate Root Theorem

100

If a polynomial has the given roots, find additional roots it must have: -2, 14 - √3

14 + √3

200

If a polynomial has the given roots, find additional roots it must have: -2i, 3 + √7

2i, 3 - √7

300

Write a polynomial function P(x) with integer coefficients which has the given root: -6i

P(x) = x² + 36

400

Write a polynomial function P(x) with integer coefficients which has the given root: 9 + 3i

P(x) = x² - 18x + 90

500

Write a polynomial function P(x) with integer coefficients which has the given roots: -3, √5

P(x) = x³ + 3x² - 5x - 15