DIVIDING POLYNOMIALS Mr. Velazquez

Honors Precalculus

FACTORING USING DIVISION

Because factoring a polynomial involves finding two multiplicative factors, we can occasionally find factors for polynomials by the use of polynomial division.

There are other techniques that can help simplifythe process of finding polynomial factors—thesewill be covered in detail in a later lecture.

Today we will be focusing on two techniques for dividing polynomials: long division and synthetic division.

POLYNOMIAL LONG DIVISION

POLYNOMIAL LONG DIVISION

Example: Dividing 6𝑥4 + 5𝑥3 + 3𝑥 − 5 by 3𝑥2 − 2𝑥

POLYNOMIAL LONG DIVISION

Example: Dividing 9𝑥2 + 6𝑥 + 5 by 3𝑥 − 2

NOTE THE

REMAINDER!

POLYNOMIAL LONG DIVISION

Example: Divide 𝑥3 + 3𝑥 − 2 by 𝑥2 + 5𝑥 − 3

THE DIVISION ALGORITHM

That last bit is somewhat important for us, as it will allow us to determine factors for polynomials that might be too difficult for conventional factoring.

In general, if 𝑓(𝑥) can be evenly divided by 𝑑(𝑥) to produce a quotient 𝑞 𝑥with no remainder, we should conclude that both 𝑑 𝑥 and 𝑞 𝑥 are factors of 𝑓(𝑥)

EXAMPLES:Divide the following using polynomial long division.

6𝑥3 − 4𝑥2 + 7

2𝑥 − 5

8𝑥4 − 3𝑥3 + 5𝑥 + 1

𝑥2 − 2𝑥 − 1

CLASSWORK, PART 1 (DO NOW!!)

On a separate sheet of paper, divide the following using polynomial long division:

SYNTHETIC DIVISION

This process looks complicated, but it actually simplifies quite easily, as we will soon see.

COMPARING METHODS

SYNTHETIC DIVISION

Here’s how we use this method to divide 5𝑥3 + 6𝑥 + 8 by 𝑥 + 2

SYNTHETIC DIVISION

Here we use it to divide 5𝑥2 + 7𝑥 − 1 by 𝑥 + 2(Note that our divisor must be a binomial of degree 1, with no coefficients)

Therefore:

EXAMPLES:Divide the following using polynomial synthetic division.

𝑥3 − 7𝑥 − 6

𝑥 + 2

THE REMAINDER THEOREM

So 𝒇 𝟐 = 𝟓

EXAMPLES:Use synthetic division and the remainder theorem to find the indicated function value.

If 𝑓 𝑥 = 6𝑥2 − 2𝑥 + 5, find 𝑓(1)

If 𝑔 𝑥 = 3𝑥3 − 5𝑥2 + 1, find 𝑔(2)

THE FACTOR THEOREM

EXAMPLES:

CLASSWORK & HOMEWORK

Use synthetic division to find the following (on the same sheet of

paper as part 1):

Homework:

2.4 – Pg. 324, #2-46

(evens)

DON’T FORGET YOUR MATH JOURNAL!