Factoring Polynomials of Higher Degree

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Factoring Polynomials of Higher Degree. Factoring Polynomials of Higher Degree. To review: What is the remainder when you divide x 3 – 4x 2 – 7x + 10 by x – 2?. Quotient. Divisor. Dividend. Remainder. Factoring Polynomials of Higher Degree. Remainder Theorem:

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### Factoring Polynomials of Higher Degree

Factoring Polynomials of Higher Degree
• To review: What is the remainder when you divide

x3 – 4x2 – 7x + 10 by x – 2?

Quotient

Divisor

Dividend

Remainder

Factoring Polynomials of Higher Degree

Remainder Theorem:

If a polynomial p(x) is divided by the

binomial x – a, the remainder

obtained is p(a).

So, looking at our example, if p(x) = x3 – 4x2 – 7x + 10 was divided by x – 2, the remainder can be determined by finding p(2).

p(x) = x3 – 4x2 – 7x + 10

p(2) = (2)3 – 4(2)2 – 7(2) + 10

= 8 – 16 – 14 + 10 = -12

Factoring Polynomials of Higher Degree

The Factor Theorem (Part 1)

If p(a) = 0, then x – a is a factor of p(x)

So let’s see what happens if p(1).

p(1) = (1)3 – 4(1)2 – 7(1) + 10

= 1 – 4 – 7 + 10

= 0

 Since p(1) = 0, we know that x – 1 is also a factor of p(x).

Factoring Polynomials of Higher Degree

Example 1: Determine the remainder when

p(x) = x3 + 5x2 – 9x – 6 is divided by x – 3.

Solution: When p(x) is divided by x – 3, the

remainder is p(3).

p(3) = 33 + 5(3)2 – 9(3) – 6

= 27 + 45 – 37 – 6

= 39

Thus, the remainder is 39. Since the remainder is

not 0, x – 3 is not a factor of p(x).

Factoring Polynomials of Higher Degree
• In order to assist with the factoring of higher order polynomials, we use a process known as synthetic division.
• A quick process to divide polynomials by binomials of the form x – a and bx – a.
Factoring Polynomials of Higher Degree
• Looking at our example:
• Synthetic Division
• Create a “L” shape
• Place “a” toward the upper left of the “L”
• Record the coefficients of the polynomials inside the “L”

2

1 -4 -7 10

Write in decreasing degree and 0’s need to be recorded as coefficients for any missing terms.

Factoring Polynomials of Higher Degree

Synthetic Division

4) Bring down 1

5) Multiply 1 by 2 and record result under – 4

7) Repeat process until you reach the last term

8) The number on the furthest right below the “L” is the remainder.

2

1 -4 -7 10

2

-4

-22

1

-2

-11

-12

Remember that 1 is the number of x2’s, -2 is the number of x’s, and -11 is the constant term of the quotient polynomial.

Factoring Polynomials of Higher Degree

Example 2: Use synthetic division to find the

quotient and remainder when x3 – 4x2 – 7x + 10 is

divided by x – 5.

5

1 -4 -7 10

5 5 -10

Remainder 0 means that x – 5 is a factor of p(x)

1 1 -2

0

Homework
• Do # 1, 3, 5, 7, 9, 17, 18, 21 on pg 130 in Section 4.3 for Monday 

Have a great weekend!!!