Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

ADDITION AND SUBTRACTION OF

POLYNOMIALS

Add polynomials.

Simplify the opposite of a polynomial.

Subtract polynomials.

Use polynomials to represent perimeter and

area.

4.4

a

b

c

d

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Objective

Add polynomials.

a

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Example A

Add. (6x3 + 7x 2) + (5x3 + 4x2 + 3)

Solution

(6x3 + 7x 2) + (5x3 + 4x2 + 3)

= (6 + 5)x3 + 4x2 + 7x + (2 + 3)

= x3 + 4x2 + 7x + 1

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Add: (3 4x + 2x2) + (6 + 8x 4x2 + 2x3)

Solution

(3 4x + 2x2) + (6 + 8x 4x2 + 2x3)

= (3 6) + (4 + 8)x + (2 4)x2 + 2x3

= 3 + 4x 2x2 + 2x3

Example B

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Add: 10x5 3x3 + 7x2 + 4 and 6x4 8x2 + 7 and

4x6 6x5 + 2x2 + 6

Solution

10x5 3x3 + 7x2 + 4

6x4 8x2 + 7

4x6 6x5 + 2x2 + 6

4x6 + 4x5 + 6x4 3x3 + x2 + 17

The answer is 4x6 + 4x5 + 6x4 3x3 + x2 + 17.

Example C

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Objective

Simplify the opposite of a polynomial.

b

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Opposites of Polynomials

To find an equivalent polynomial for the

opposite, or additive inverse, of a polynomial,

change the sign of every term. This is the same

as multiplying by 1.

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Simplify: (8x4 x3 + 9x2 2x + 72)

Solution

(8x4 x3 + 9x2 2x + 72)

= 8x4 + x3 9x2 + 2x 72

Example D

34

34

34

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Objective

Subtract polynomials.

c

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Subtraction of Polynomials

We can now subtract one polynomial from another by

adding the opposite of the polynomial being

subtracted.

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(10x5 + 2x3 3x2 + 5) (3x5 + 2x4 5x3 4x2)

Solution

(10x5 + 2x3 3x2 + 5) (3x5 + 2x4 5x3 4x2)

= 10x5 + 2x3 3x2 + 5 + 3x5 2x4 + 5x3 + 4x2

= 13x5 2x4 + 7x3 + x2 + 5

Example E Subtract:

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Subtract: (8x5 + 2x3 10x) (4x5 5x3 + 6)

Solution

(8x5 + 2x3 10x) (4x5 5x3 + 6)

= 8x5 + 2x3 10x + (4x5) + 5x3 6

= 4x5 + 7x3 10x 6

Example F

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Write in columns and subtract:

(6x2 4x + 7) (10x2 6x 4)

Solution

6x2 4x + 7

(10x2 6x 4)

4x2 + 2x + 11

Example G

Remember to change

the signs

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Objective

Use polynomials to represent

perimeter and area.

d

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A 6-ft by 5-ft hot tub is installed on an outdoor

deck measuring w ft by w ft. Find a polynomial

for the remaining area of the deck.

Solution

1. Familiarize. We make a drawing of the

situation as follows.

Example H

w ft

w ft

7 ft

5 ft

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2. Translate.

Rewording: Area of Area of Area

deck tub = left over

Translating: w ft w ft 5 ft 7 ft = Area left over

3. Carry out.

w2 ft2 35 ft2 = Area left over.

4. Check. As a partial check, note that the units in the

answer are square feet, a measure of area, as

expected.

5. State. The remaining area in the yard is (w2 35)ft2.

continued

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Section 4.4

1. Add: (6x5 – x3 + 4x2 + x) +

(x4 – x3 + 8x2 – x + 2).

a) 7x5 – 2x3 + 12x2 + 2x + 2

b) 6x5 + x4 – 2x3 + 12x2 + 2

c) 7x5 – 2x3 + 12x2 + 2

d) 6x5 + x4 + 12x2 + 2

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Section 4.4

1. Add: (6x5 – x3 + 4x2 + x) +

(x4 – x3 + 8x2 – x + 2).

a) 7x5 – 2x3 + 12x2 + 2x + 2

b) 6x5 + x4 – 2x3 + 12x2 + 2

c) 7x5 – 2x3 + 12x2 + 2

d) 6x5 + x4 + 12x2 + 2

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Section 4.4

2. Subtract: (1.1x3 – 8x2 + 4.5x) –

(3x3 – x2 + 0.5x – 6).

a) –2.2x3 – 7x2 + 4x + 6

b) –1.9x3 – 7x2 + 4x + 6

c) –2.9x3 – 7x2 + 4x + 6

d) –1.9x3 – 9x2 + 4x + 6

Slide 4- 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Section 4.4

2. Subtract: (1.1x3 – 8x2 + 4.5x) –

(3x3 – x2 + 0.5x – 6).

a) –2.2x3 – 7x2 + 4x + 6

b) –1.9x3 – 7x2 + 4x + 6

c) –2.9x3 – 7x2 + 4x + 6

d) –1.9x3 – 9x2 + 4x + 6