3.2 Polynomials-- Properties of Division

Leading to Synthetic Division

Part 1Some serious relationships

Start with a POD

Using long division,

divide x4 - 16 by x2 + 3x + 1

By hand.

On CAS.

Start with a POD

Using long division,

divide x4 - 16 by x2 + 3x + 1

Quotient: x2 - 3x + 8

Remainder: -21x - 24

The POD leads to The Division Algorithm for Polynomials:

f(x) = d(x) · q(x) + r(x)

where d, q, and r are polynomials,

d is the divisor, q is the quotient, r is the remainder,

and r(x) has a smaller degree than d(x).

There’s a pattern– what is it?Find the remainders with LINEAR divisors:

Compare that to f(3)– the top polynomial.

Compare that to f(-4).

3

842 23

x

xxx 8423 23 xxxx

4

932 234

x

xxxx9324 234 xxxxx

The Relationship The Division Algorithm for Polynomials:

f(x) = d(x) · q(x) + r(x)

What happens when the remainder = 0?

The Factor Theorem:

If the remainder = 0, then f(c) = 0.

If f(c) = 0, then c is a root/ solution/ zero of f(x) and (x - c) is a factor of f(x).

The ProofStart with: f(x) = d(x) · q(x) + r(x)

What if d(x) = x - c ? (This is a linear divisor.) f(x) = (x - c)q(x) + r(x)

r(x) must be of a degree less than (x - c), so r(x) must be a constant, k

Let’s see what happens when we plug c in for x to find f(c):

f(c) = (c - c)q(c) + k = 0·q(c) + k

In other words, f(c) is the remainder k.

The Remainder Theorem:

If polynomial f(x) is divided by x - c, the constant remainder r(x) is f(c).

Use it 1. If f(x) = x3 - 3x2 + x + 5, use the Remainder Theorem to find

f(-4).

2. Show that (x - 2) is a factor of f(x) = x3 - 4x2 + 3x + 2.

3. Find a polynomial f(x) that has zeros at x = 2, -1, and 3.

(How many could you find? How many that are 3rd degree?

Use it 1. If f(x) = x3 - 3x2 + x + 5, use the Remainder Theorem to find

f(-4). Divide f(x) by (x+4).

2. Show that (x - 2) is a factor of f(x) = x3 - 4x2 + 3x + 2.Show that f(2) = 0.

3. Find a polynomial f(x) that has zeros at x = 2, -1, and 3. f(x) = (x-2)(x+1)(x-3)

(How many could you find? How many that are 3rd degree? The answer to both of these is “infinite.” Consider f(x) = (x-2)(x+1)(x-3)2 to illustrate the first answer, and f(x) = 3(x-2)(x+1)(x-3) to illustrate the second answer.)

Part 2Let’s make it easier.

Dividing polynomials

Use long division (not the easy part):

x 3 2x4 5x3 2x 8

Dividing polynomials

Use long division:

With a remainder of 25.

What is f(-3)?

x 3 2x4 5x3 2x 82x3 x2 3x 11

Dividing polynomials

Now, use synthetic division:

becomes -3 |2 5 0 -2 -8

|____________

What is the remainder?

What is the quotient?

x 3 2x4 5x3 2x 82x3 x2 3x 11

Use this to find zeros of polynomials

What polynomials are represented below?

What is the remainder? The quotient?

What is the full factorization over the set of integers?

What does it all mean?

-11 | 1 8 -29 44

|____________

Use this to find zeros of polynomials

What polynomials are represented below?What is the remainder? The quotient? 0 x2 - 3x + 4What does it all mean?

For f(x) = x3 + 8x2 - 29x +44

-11 is a zero/ solution/ root(x + 11) is a factor of f(x)

What are the integer factors of f(x)? (x + 11)(x2 - 3x + 4)

Use this to find other information about polynomials

For f(x) = x3 + 8x2 - 29x +44

use synthetic division to find f(2).

Use this to find other information about polynomials

For f(x) = 2x4 + 5x3 - 3k,

What is k if f(2) = 39?

Summary (using different letters)If f(a) = b, then

1. b is the remainder of f(x)/(x - a).2. (a, b) is on the graph of f(x).3. The value of f(x) at a is b.

If b = 0 also, then

1. (x - a) is a factor of f(x). Note: linear factor!2. a is a zero of f(x).3. (a, 0) is on the graph of f(x)4. a is an intercept of f(x).5. a is a solution of f(x) = 0.

Use it one more time1. Give a factor of f(x) = x3 - 5x2 - 6x if (6, 0) is on the graph.

What is another factor?

2. If (-4, k) is on the graph of x3 - 2x + 2, what is k? How many ways could you corroborate your answer?