14-3 Adding Polynomials

Warm UpCombine like terms.

1. 9x + 4x 2. –3y + 7y3. 7n + (–8n) + 12n

Find the perimeter of each rectangle.

4. a 10 ft by 12 ft rectangle5. a 5 m by 8 m rectangle

Simplify.

6. 3(2x2 – x) + x2+ 1

13x 4y11n

44 ft26 m

7x2 – 3x + 1

14-3 Adding Polynomials

Learn to add and subtract polynomials.

14-3 Adding Polynomials

Example 1A: Adding Polynomials Horizontally

Add.

(5x3 + x2 + 2) + (4x3 + 6x2)

(5x + x + 2) + (4x + 6x )3 2 3 2

5x + x + 2 + 4x + 6x 3 2 3 2

9x + 7x + 23 2

Associative Property

Combine like terms.

14-3 Adding Polynomials

Example 1B: Adding Polynomials Horizontally

Add.

(6x3+ 8y2 + 5xy) + (4xy – 2y2)

(6x3+ 8y2+ 5xy) + (4xy – 2y2)

6x3 + 8y2 + 5xy + 4xy – 2y2

6x + 6y + 9xy3 2

Associative Property

Combine like terms.

14-3 Adding Polynomials

Example 1C: Adding Polynomials Horizontally

Add.

(3x2y – 5x) + (4x + 7) + 6x2y

(3x2y – 5x) + (4x + 7) + 6x2y

9x2y – x + 7

Associative Property

Combine like terms.

3x2y – 5x + 4x + 7 + 6x2y

14-3 Adding Polynomials

Example 2A

Add.

(3y4 + y2 + 6) + (5y4 + 2y2)

(3y + y + 6) + (5y + 2y )4 2 4 2

3y + y + 6 + 5y + 2y 4 2 4 2

8y + 3y + 64 2

Associative Property

Combine like terms.

14-3 Adding Polynomials

Example 2B

Add.

(9x3 + 6p2 + 3xy) + (8xy – 3p2)

(9x + 6p2 + 3xy) + (8xy – 3p2)3

9x + 6p + 3xy + 8xy – 3p 3 2 2

9x3+ 3p2 + 11xy

Associative Property

Combine like terms.

14-3 Adding Polynomials

Example 2C

Add.

(3z2w – 5x) + (2x + 8) + 6z2w

(3z2w – 5x) + (2x + 8) + 6z2w

9z2w – 3x + 8

Associative Property

Combine like terms.

3z2w – 5x + 2x + 8 + 6z2w

14-3 Adding Polynomials

You can also add polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms. If the terms are rearranged, remember to keep the correct sign with each term.

14-3 Adding Polynomials

Example 3: Adding Polynomials Vertically

Add.

A. (4x2 + 2x + 11) + (2x2 + 6x + 9)

4x2 + 2x + 11

+ 2x2 + 6x + 9

6x2 + 8x + 20 Combine like terms.

Place like terms in columns.

14-3 Adding Polynomials

Example 3: Adding Polynomials Vertically

Add.

B. (3mn2 – 6m + 6n) + (5mn2 + 2m – n)

C. (–x2y2 + 5x2) + (–2y2 + 2) + (x2 + 8)

+ 5mn2 + 2m – n

8mn2 – 4m + 5n

–2y2 + 2

+ x2 + 8

–x2y2 + 6x2 – 2y2 + 10

3mn2 – 6m + 6n

–x2y2 + 5x2

Combine like terms.

Place like terms in columns.

Combine like terms.

Place like terms in columns.

14-3 Adding Polynomials

Example 4

Add.

A. (6x2 + 6x + 13) + (3x2+ 2x + 4)

6x2 + 6x + 13

+ 3x2 + 2x + 4

9x2 + 8x + 17 Combine like terms.

Place like terms in columns.

14-3 Adding Polynomials

Example 4

Add.

B. (4mn2 + 6m + 2n) + (2mn2 – 2m – 2n)

C. (x2y2 – 5x2) + (2y2 – 2) + (x2)

+ 2mn2 – 2m – 2n

6mn2 + 4m

4mn2 + 6m + 2n

Combine like terms.

Place like terms in columns.

2y2 – 2

+ x2

x2y2 – 4x2 + 2y2 – 2

x2y2 – 5x2

Combine like terms.

Place like terms in columns.

14-3 Adding Polynomials

Subtraction is the opposite of addition. To subtract a polynomial, you need to find itsopposite.

14-3 Adding Polynomials

Find the opposite of each polynomial.

Example 1: Finding the Opposite of a Polynomial

A. 8x3y4z2

–(8x3y4z2)

–8x3y4z2 Distributive Property.

B. –3x4 + 8x2

–(–3x4 + 8x2)

3x4 – 8x2 Distributive Property.

14-3 Adding Polynomials

Find the opposite of the polynomial.

Additional Example 1: Finding the Opposite of a Polynomial

C. 9a6b4 + a4b2 – 1

–(9a6b4 + a4b2 – 1)

–9a6b4 – a4b2 + 1 Distributive Property.

14-3 Adding Polynomials

To subtract a polynomial, add its opposite.

14-3 Adding Polynomials

Subtract.

Example 1: Subtracting Polynomials Horizontally

A. (5x2 + 2x – 3) – (3x2 + 8x – 4)

= (5x2 + 2x – 3) + (–3x2 – 8x + 4)

= 5x2 + 2x – 3 – 3x2 – 8x + 4

Add theopposite.

Associativeproperty.

= 2x2 – 6x + 1 Combine liketerms.

14-3 Adding Polynomials

Subtract.

Example 1: Subtracting Polynomials Horizontally

B. (b2 + 4b – 1) – (7b2 – b – 1)

= (b2 + 4b – 1) + (–7b2 + b + 1)

= b2 + 4b – 1 – 7b2 + b + 1

= –6b2 + 5b

Add the opposite.

Associative property.

Combine like terms.

14-3 Adding Polynomials

Example 2A

Subtract.

(2y3 + 3y + 5) – (4y3 + 3y + 5)

Add theopposite.

Associativeproperty.

= –2y3 Combine liketerms.

= (2y3 + 3y + 5) + (–4y3 – 3y – 5)

= 2y3 + 3y + 5 – 4y3 – 3y – 5

14-3 Adding Polynomials

Example 2B

Subtract.

(c3 + 2c2 + 3) – (4c3 – c2 – 1)

= –3c3 + 3c2 + 4

Add the opposite.

Associative property.

Combine like terms.

= (c3 + 2c2 + 3) + (–4c3 + c2 + 1)

= c3 + 2c2 + 3 – 4c3 + c2 + 1

14-3 Adding Polynomials

You can also subtract polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms.

14-3 Adding Polynomials

Subtract.

Example 3: Subtracting Polynomials Vertically

(2n2 – 4n + 9) – (6n2 – 7n + 5)

(2n2 – 4n + 9) 2n2 – 4n + 9

– (6n2 – 7n + 5) + –6n2 + 7n – 5

–4n2 + 3n + 4

Add the opposite.

14-3 Adding Polynomials

Subtract.

Example 4: Subtracting Polynomials Vertically

(10x2 + 2x – 7) – (x2 + 5x + 1)

(10x2 + 2x – 7) 10x2 + 2x – 7

– (x2 + 5x + 1) + –x2 – 5x – 1

9x2 – 3x – 8

Add the opposite.

14-3 Adding Polynomials

Subtract.

Example 5: Subtracting Polynomials Vertically

(6a4 – 3a2 – 8) – (–2a4 + 7)

(6a4 – 3a2 – 8) 6a4 – 3a2 – 8– (–2a4 + 7) + 2a4 – 7

8a4 – 3a2 – 15

Rearrange as needed.

14-3 Adding Polynomials

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

14-3 Adding Polynomials

Lesson Quiz: Part I

Add.

1. (2m2 – 3m + 7) + (7m2 – 1)

2. (yz2 + 5yz + 7) + (2yz2 – yz)

3.

9m2 – 3m + 6

3yz2+ 4yz + 7

7xy + 2x + 3y + 22

(2xy2 + 2x – 6) + (5xy2 + 3y + 8)

14-3 Adding Polynomials

Lesson Quiz

Find the opposite of each polynomial.

Subtract.

3. (3z2 – 7z + 6) – (2z2 + z – 12)

3m3 – 2m2n

z2 – 8z + 18

–3a2b2c3 2. –3m3 + 2m2n

4. –18h3 – (4h3 + h2 – 12h + 2)

5. (3b2c + 5bc2 – 8b2)

– (4b2c + 2bc2 – c2)

– b2c + 3bc2 – 8b2 + c2

– 22h3 – h2 + 12h – 2

1. 3a2b2c3

14-3 Adding Polynomials

1. Add (4p2 – 8p +11) + (6p2 – 9).

A. 10p2 – 8p + 2

B. 10p2 + 8p + 20

C. 2p2 + 8p + 2

D. 10p2 – 8p + 20

Lesson Quiz for Student Response Systems

14-3 Adding Polynomials

2. Add (gh2 + 9gh + 11) + (3gh2 – gh).

A. 2gh2 + 8gh + 11

B. 2gh2 + 10gh + 11

C. 4gh2 + 8gh + 11

D. 4gh2 + 10gh + 11

Lesson Quiz for Student Response Systems

14-3 Adding Polynomials

3. Add (7uv3 + 11u) + (6uv3 – u – 9) + (4u – 2).

A. 13uv3 – 6u – 7

B. uv2 – 16u + 7

C. uv3 + 14u – 11

D. 13uv3 + 14u – 11

Lesson Quiz for Student Response Systems

14-3 Adding Polynomials

3. Subtract.(11p2 – 5p + 9) – (3p2 + p – 17)

A. 14p2 + 4p – 8

B. 14p2 – 6p + 26

C. 8p2 – 6p + 26

D. 8p2 + 4p – 8

Lesson Quiz for Student Response Systems