Roots and Factors of Polynomials Part 2

​Consider the following factors:
(x - 4) (x - 4)² (x² - 4)


Graph each of them in Desmos on the next slide and describe what they have in common and what are the differences.

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Match

Match the factor with its description:

• (x - 4)

• (x - 4)²

• (x² - 4)

Single root at 4

double root at 4

2 different roots: 2 and -2

• Single root at 4

• double root at 4

• 2 different roots: 2 and -2

​Now consider the following factors: Graph them in Desmos


(x - 5) (x - 5)² (x² - 5)

Match

Match the factors and descriptions

• (x - 5)

• (x² - 5)

• (x - 5)²

one root at 5

2 irrational roots (positive & negative)

double root at 5

• one root at 5

• 2 irrational roots (positive & negative)

• double root at 5

​Now try without the graphing....

Categorize

Options (9)

(x + 3)

(x - 1)²

(x² - 2)

(x² - 11)

(x² + 25)

(x - 5)

(x + 22)

(x + 2)²

(x² - 6)

Put the factors in the correct categories:

(carefully think about what the previous factors looked like)

Single Root

Double Root

2 irrational Roots

None of those

​On the next slide....
Try graphing a 3rd degree polynomial in standard form with 4 terms.
Try making it have 3 integer roots.

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Factors (x² - number)

​When the number is not a perfect square

set the factor = 0 and then solve for x

Ex: x² - 5 = 0 so x² = 5 square root both sides

x = +√5 and x = -√5 These are called conjugate irrational roots
**irrational roots ALWAYS come in pairs!!

Math Response

(x² - 13)

one of the roots is x=√13

So what is the other root?

(include x= in your answer)

Type answer here

Correct Answer: x=-\sqrt{13}

Deg^°

Rad

Math Response

The factor is (x² - 27)

One of the roots is x= -√27

What is the other root?

Type answer here

Correct Answer: x=\sqrt{27}

Deg^°

Rad

Something to think about...

​What if the factor is (x² + 9)?

What happens when you set the factor = to ZERO?
When you try to solve for x, what happens?

This is a future lesson, but you already have a head- start!!