Factoring Polynomial Functions

How do you find the degree of a polynomial in standard form?

Look at the highest exponent in the equation, like x^4 - 4x² (degree is 4)

Imaginary vs Real Roots

Real roots: The ones that touch the x-axis
Imaginary: Root’s that should exist by the degree, but don’t show on x-axis

The degree of a polynomial gives the number of _____

solutions

How must complex roots be found?

Factoring

Extending Difference of Squares (Two Terms):

a² - b² = (a-b)(a+b)

Example of Difference of Squares:

a^4 - b^4

(a² + b²)(a²-b²)

Factor: 9x² - 4

(3x-2)(3x+2)

Factor: x^4 - 1

(x² - 1) (x² + 1)

Steps to Factoring

1. Take Out GCF

2. Find the common multiple with grouping

Difference of Cubes

a³ - b³ = (a - b)(a² + ab + b²)

Factor: 27x³ + 8 (difference of cubes)

(3x + 2) ((3x)² + 3x (2) + (2)²) = (3x+2) (9x² - 6x + 4)

Factor: 64x³ - 125

(4x - 5) (16x² + 20x + 25)

Difference of cubes negative sign in front of b

The - (negative) is not b, it’s just apart of the equation. You ignore with multiply ab in difference of cubes

How to divide polynomials

Use long division and match the divisor with the dividend.

Divide:

6x³ + 7x² - 16x + 10
—————————-
3x² - 4x + 2

3x² - 4x + 2

Sum of Cubes

a³ - b³ = (a - b)(a² - ab + b²)

Difference between “Sum of Cubes/Squares and Difference of Cubes/Squares”

Sum is used for equations where the squares/cubes are being added: 3x² + 27

Difference is for subtraction: 3x² - 27

Sum of Squares

a² + b² ( a+ bi) (a - bi)