6.5 The Remainder and Factor Theorems p. 352

How do you divide polynomials?What is the remainder theorem?

What is the difference between synthetic substitution and synthetic division?

What is the factor theorem?What is a zero of a polynomial function?

When you divide a Polynomial f(x) by a divisor d(x), you get a quotient polynomial q(x) with a remainder r(x) written:

f(x) = q(x) + r(x)d(x) d(x)

The degree of the remainder must be less than the degree of the divisor!

Polynomial Long Division:

You write the division problem in the same format you would use for numbers. If a term is missing in standard form …fill it in with a 0 coefficient.

Example: 2x4 + 3x3 + 5x – 1 = x2 – 2x + 2

1503222 2342 xxxxxx

2x4 = 2x2

x2

2x2

1503222 2342 xxxxxx

2x2

+4x2-4x32x4-( )

- 4x27x3 +5x

7x3 = 7x x2

+7x

7x3 - 14x2 +14x-( )

10x2 - 9x -1

+10

10x2 - 20x +20-( )

11x - 21

remainder

The answer is written:

2x2 + 7x + 10 + 11x – 21 x2 – 2x + 2

Quotient + Remainder over divisor

Now you try one!

y4 + 2y2 – y + 5 =y2 – y + 1

Answer: y2 + y + 2 + 3 y2 – y + 1

Remainder Theorem:

If a polynomial f(x) is divisible by (x – k), then the remainder is r = f(k).

Now you will use synthetic division (like synthetic substitution)

f(x)= 3x3 – 2x2 + 2x – 5 Divide by x - 2

f(x)= 3x3 – 2x2 + 2x – 5 Divide by x - 2 Long division results in ?...... 3x2 + 4x + 10 + 15

x – 2 Synthetic Division: f(2) = 3 -2 2 -5

23

6

4

8

10

20

15

Which gives you: 3x2 + 4x + 10 + 15 x-2

Synthetic Division Practice 1

Divide x3 + 2x2 – 6x -9 by (a) x-2 (b) x+3 (a) x-2 1 2 -6 -9

2 1

2

4

8

2

4

-5

Which is x2 + 4x + 2 + -5

x-2

Synthetic Division Practice cont.

(b) x+3 12 -6 -9 -

31

-3

-1

3

-3

9

0

x2 – x - 3

Factor Theorem:

A polynomial f(x) has factor x-k if f(k)=0

note that k is a ZERO of the function because f(k)=0

Factoring a polynomial

Factor f(x) = 2x3 + 11x2 + 18x + 9 Given f(-3)=0

Since f(-3)=0 x-(-3) or x+3 is a factor So use synthetic division to find the

others!!

Factoring a polynomial cont.

211 18 9

-3 2

-6

5

-15

3

-9

0

(x + 3)(2x2 + 5x + 3)

So…. 2x3 + 11x2 + 18x + 9 factors to:

Now keep factoring-- gives you:

(x+3)(2x+3)(x+1)

Your turn!

Factor f(x)= 3x3 + 13x2 + 2x -8 given f(-4)=0

(x + 1)(3x – 2)(x + 4)

Finding the zeros of a polynomial function f(x) = x3 – 2x2 – 9x +18. One zero of f(x) is x=2 Find the others! Use synthetic div. to reduce the degree

of the polynomial function and factor completely.

(x-2)(x2-9) = (x-2)(x+3)(x-3) Therefore, the zeros are x=2,3,-3!!!

Your turn!

f(x) = x3 + 6x2 + 3x -10 X=-5 is one zero, find the others!

The zeros are x=2,-1,-5 Because the factors are (x-2)(x+1)(x+5)

How do you divide polynomials?

By long division What is the remainder theorem?

If a polynomial f(x) is divisible by (x – k), then the remainder is r = f(k).

What is the difference between synthetic substitution and synthetic division?

It is the same thing What is the factor theorem?

If there is no remainder, it is a factor. What is a zero of a polynomial function?

One of the answers

AssignmentPage 356, 15-53 odd