Polynomials and Partial Fractions

Objectives

In this lesson, you will learn how to perform arithmetic operations on polynomials.

4.1 Polynomials

2P 5 3 1x x x

The following expressions are polynomials.

3 23 4 2 1x x x

4 2127 3x x We also use this

form of function notation to denote

a polynomial.7 is the coefficient of x4 and

½ is the coefficient of x2.

A polynomial in a variable x, is a sum of terms, each of the form axn, where a is a constant and n is a non-negative integer.

Polynomials and Partial Fractions

2P 3 3 3 1

2 21 4 6 4x x x x

2 2P 2Q 1 2 2 3 2x x x x x x

25 5 5x x

Substitute for x in P(x).

Combine the two polynomial functions.

If P(x) = x2 + x + 1 and Q(x) = 2x2 – 3x + 2, find

Polynomials and Partial Fractions

Example

(a) P(3),(b) P(x) + 2Q(x).

13

25 1x x

Polynomials and Partial Fractions

In this lesson, you will learn how to find unknown constants in a polynomial identity.

4.2 Identities

Objectives

2 22 4 4x x x

An expression involving polynomials that can be solved to find a specific value for x, is an equation.

2 2 4 3x x x 2 2 3 0x x

This is always true, so, it is an identity.

We have solved for x, so, this is an equation.

Equations and Identities

Polynomials and Partial Fractions

1 and 3x x

An expression involving polynomials that is true for all values of x is an identity.

3 22 6 5 2 2 2x x x x ax b

6 2 2a If x = 2, then

(x – 2) = 0.

The coefficients of x2 could

be used too.

Find the values of a and b in the following identity.

Polynomials and Partial Fractions

16 12 5 b 2 8a 3

2 2 6 2 5 b

9b

Let x = 2. Equate the coefficients of x.

4a

x-3 -2 -1 1 2 3

y

-4

-2

2

4

6

8

10

y = 2x3

– 6x + 5

y = (x – 2)(2x2

+ 4x + 2) + 9

Check the results

graphically.

Example

23 2 23 2 16 1 2 1 2x x x Ax x B x x C x

If x = 1, then(x – 1) = 0.

Find the values of A, B and C in the following identity.

Polynomials and Partial Fractions

18 3 6C C Let x = 1.

Let x = 2.

3 21 3 1 2 1 16 1 2C

If x = 2, then

(x – 2) = 0.

3 2 22 3 2 2 2 16 2 2 1 6 2 2A

8 12 4 16 4 24A 4 8 2A A

Equate the coefficients of x3.

1 A B 1B

Example

Polynomials and Partial Fractions

In this lesson, you will learn how to divide one polynomial by another.

4.3 Dividing Polynomials

Objectives

8 1327Subtract 1 × 8 from

13.

Divide 13 by 8.

A reminder about long division of integers.

Polynomials and Partial Fractions

1 8 1327

8

52

16 8 1327

48

47

1658 1327

40

7

8

5 48

4

Bring the 2 down.

Divide 52 by 8.

Subtract 6 × 8 from 52.

Bring the 7 down.

Divide 47 by 8.

Subtract 5 × 8 from 47.

divisordividend

quotient

remainder

Therefore 1327 8 165 7

We will now apply the same process to

polynomials.

For any division,

Polynomials and Partial Fractions

dividend = divisor × quotient + remainder

or

dividend ÷ divisor = quotient +divisor

remainder

4 3 21 3 2 7x x x x x Subtract x3 × (x – 1)

from x4 + 3x3.

Divide x4 by x.

This is the same method as long division with integers.

Polynomials and Partial Fractions

3

4 3 2

1 3 2 7

xx x x x x

4 3

3 2

4 2

x x

x x

3 2

4 3 2

4 1 3 2 7

x xx x x x x

3 2

2

4 4

2

x x

x x

3 2

4 3 2

4 2 1 3 2 7

x x xx x x x x

22 2

3

x x

x

4 3

3 4

x x

x

3 2

2

4 4

2

x x

x

Bring the – 2x2

down.

Divide 4x3 by x.

Subtract 4x2× (x – 1) from 4x3 –

2x2.

Bring the x down.

Divide 2x2 by x.

Subtract 2x× (x – 1) from 2x2 + x.

divisor dividend

quotient

remainder

22 2

3 7

x x

x

3 3

4

x

Divide 3x by x.

Subtract 3× (x – 1) from 3x –

7.

3 2

4 3 2

4 2 31 3 2 7

x x xx x x x x

Bring the – 7 down.

The following identity is always true

Polynomials and Partial Fractions

3 2

4 3 2

4 2 31 3 2 7

x x xx x x x x

dividend = divisor × quotient + remainder

+ remainderquotient

divisor dividend

4

4 3 2 3 23 2 7 1 4 2 3 4x x x x x x x x

Therefore

Example

Divide .3 22 4 2 by 2x x x x

3 22 2 4 2x x x x Subtract 2x2 × (x – 2) from 2x3 – 4x2.

Divide 2x3 by x.

Polynomials and Partial Fractions

3 22 4

0

x x

x

2

3 2

2 12 2 4 2

xx x x x

2

0

x

2

3 2

2 2 2 4 2

xx x x x

3 22 4

0

x x

Bring the x down.

Divide 0 by x.

There is no x term.

Bring the – 2 down.

Divide x by x.Subtract 1× (x – 2)

from x – 2.

2

3 2

2 2 2 4 2

xx x x x

3 22 4

2

x x

x

3 2 2Hence 2 4 2 2 2 1x x x x x

Example

Divide .4 3 2 24 5 2 by 2x x x x x

2 4 3 22 4 5 2x x x x x

Subtract 4x2 × (x2 – x + 2)

from 4x4 – 5x3 + x2

Divide 4x4 by x2.

Polynomials and Partial Fractions

4 3 2

3 2

4 4 8

7

x x x

x x

2

2 4 3 2

4 8

2 4 5 2

x x

x x x x x

28 8 16

6 14

x x

x

2

2 4 3 2

4

2 4 5 2

x x

x x x x x

4 3 2

3 2

4 4 8

7

x x x

x x

No term to bring down.

Divide –x3 by x2.

Subtract –x × (x2 – x + 2) from – x3 – 7x2.

Bring the – 2 down.

Subtract – 8 × (x2 – x + 2) from – 8 x2 +8x –

2.

2

2 4 3 2

4

2 4 5 2

x

x x x x x

3 2

2

2

8 2

x x x

x x

4 3 2 2 2Hence 4 5 2 2 4 8 6 14x x x x x x x x

3 2

2

2

8 2 2

x x x

x x

Divide –8x2 by x2.