1. Factoring Polynomials of Higher Degree

2. 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

3. 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 – = -12

4. 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 –
= 0
 Since p(1) = 0, we know that x – 1 is also a factor of p(x).

5. 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) = (3)2 – 9(3) – 6
= – 37 – 6
= 39
Thus, the remainder is 39. Since the remainder is
not 0, x – 3 is not a factor of p(x).

6. 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.

7. 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
Write in decreasing degree and 0’s need to be recorded as coefficients for any missing terms.

8. 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 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.

9. 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
Remainder 0 means that x – 5 is a factor of p(x)

10. Homework
Do # 1, 3, 5, 7, 9, 17, 18, 21 on pg 130 in Section 4.3 for Monday 
Have a great weekend!!!