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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


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factoring polynomials of higher degree1
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 degree2
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 degree3
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 degree4
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 degree5
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 degree6
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 degree7
Factoring Polynomials of Higher Degree

Synthetic Division

4) Bring down 1

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

6) Add -4 to 2 and record answer below the “L”

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 degree8
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
Homework
  • Do # 1, 3, 5, 7, 9, 17, 18, 21 on pg 130 in Section 4.3 for Monday 

Have a great weekend!!!