Factoring Polynomials of Higher Degree

The Factor Theorem Part II

Factoring Polynomials of Higher Degree

From last class, synthetic division is a

quick process to divide polynomials by

binomials of the form x – a and bx – a.

2 1 -4 -7 10

1

2

-2

-4

-11 -12

-22

Factoring Polynomials of Higher Degree

Example 1: Use synthetic division to find the

quotient and remainder when 6x3 – 25x2 – 29x + 25

is divided by 2x – 1.

Solution:

b

a

6 -25 -29 25

6

3

-22

-11

-40 5

-20bx – a is written

in the form in

front of the “L”.

1

2

Factoring Polynomials of Higher Degree

The Factor Theorem (Part 2)

If p(x) = anxn + an-1xn-1 + an-2xn-2 +…+ axx2 + azx + a0 is a

polynomial function with integer coefficients, and if bx – a

is a factor of p(x) where “b” and “a” are also integers, then

“b” is a factor of the leading coefficient an and “a” is a

factor of the constant term a0.

“b” and “a” should have no factors in common.

Factoring Polynomials of Higher Degree

Example 2: Factor x3 + 5x2 + 2x – 8.

Solution: If x3 + 5x2 + 2x – 8 has a factor of

the form bx – a, then “a” is a factor of -8 and

“b” is a factor of 1.

A list of possible factors is x – 1, x – 2, x – 4,

x – 8, x + 1, x + 2, x + 4, x + 8.

We will use synthetic division to try and find one factor. Start by testing x – 1.

Factoring Polynomials of Higher Degree

1 5 2 -8

1

1

6

6

8 0

8

1

We note that x – 1 must be a factor since the remainder is 0.

Thus, we have:

x3 + 5x2 + 2x – 8 = (x – 1)(x2 + 6x + 8)

= (x – 1)(x + 2)(x +4)

Factor the trinomial

Factoring Polynomials of Higher Degree

Note that only 3 of the possible 8 solutions for the polynomial were factors.

It is unlikely that you will actually find a factor on your first try.

It will take a few attempts to find the first one, and then factor the remaining quadratic by inspection.

SummaryTo factor a polynomial of higher degree:

1) Write the polynomial in decreasing degree inserting 0’s for any missing terms.

2) Make a list of all possible factors bx – a, where “a” is a factor of the constant term and “b” is a factor of the leading coefficent. “b” and “a” should have no factors in common.

3) Test the list of potential factors using synthetic division.

4) When you find a factor, factor the remaining quadratic.

Factoring Polynomials of Higher Degree

Homework

Do # 19, 25, 29, 31, 33, and 35 on page 130 from section 4.3 for Monday April 20th

Have a Safe and Fun Easter Break