Polynomials

Definitions

100

What is the degree of a polynomial?

monomial with the highest exponent

200

What is a prime polynomial?

A polynomial that cannot be factored

300

What is the lead coefficient?

number in front of the variable with the highest exponent

400

When do relative minimums or maximums occur?

when the graph changes from increasing to decreasing or decreasing to increasing

500

What is the Location Principle?

The Location Principle states that if the value of f(x) changes sign from one value of x to the next, then there is a zero in between those two x-values. 

Operations

100

Simplify (2x⁴)⁶

64x²⁴

200

Simplify 8x(2y)³

64xy³

300

Simplify (3x²+1) + (8x²-8)

11x² - 7

400

Simplify (w + 2y)(w² - 2wy + 4y²)

w³ + 8y³

500

Simplify (4d²t⁵v^-4)(-5dt^-3v^-1)

(-20d³t²)/(v⁵)

Dividing

100

Simplify (15x¹⁰ - 5x⁸) / 5x⁴

3x⁶ - r⁴

200

Simplify (4a³ - 8a² + 4a) / (4a)

a² - 2a + 1

300

Simplify (2x² + 3x - 14) / (x - 2)

2x + 7

400

Simplify (x³ - 64) / (x - 4)

x² + 4a + 16

500

Simplify (x⁴ - 3x³ + 5x - 6) / (x + 2)

x³ - 5x² + 10x - 15  Remainder: 24

Graphing

100

What is domain and range?

domain: x values

range: y values

200

What does the graph of an even polynomial look like?

Ends of graph are both going up or both going down

300

How many zeroes does the graph of the following polynomial have? x³ - 4x² + 2

3 zeroes 

400

State the degree and leading coefficient of 7x³-4x²+x.

Degree: odd, LC: positive

500

Write the end behavior for the polynomial f(x) = x³ - 12x² + 2x - 4.

as f(x) --> -8, x --> -8

as f(x) --> 8, x --> 8

Factoring

100

Factor 10x²+5x

5x(2x + 1)

200

Factor x² - 12x + 32 

(x - 4)(x - 8)

300

Factor x³ + 8

(x + 2)(x² - 2x + 4)

400

Prove that 4(x - 5)(x - 10) = 4x² - 60x + 200 is an identity. 

4(x - 5)(x - 10) = 4x² - 60x + 200

500

Solve x⁴ - 29x² + 100 = 0.

x = -2, 2, -5, 5