Warm Up Combine like terms. 1. 9x + 4x 2. –3y + 7y 3. 7n + (–8n) + 12n Find the perimeter of each rectangle. 4. a 10 ft by 12 ft rectangle 5. a 5 m by 8 m rectangle Simplify. 6. 3(2x2 – x) + x2+ 1 13x 4y 11n 44 ft 26 m 7x2 – 3x + 1

Learn to add polynomials.

Example 1A: Adding Polynomials Horizontally (5x3 + x2 + 2) + (4x3 + 6x2) (5x + x + 2) + (4x + 6x ) 3 2 5x + x + 2 + 4x + 6x 3 2 Associative Property 9x + 7x + 2 3 2 Combine like terms.

Example 1B: Adding Polynomials Horizontally (6x3+ 8y2 + 5xy) + (4xy – 2y2) (6x3+ 8y2+ 5xy) + (4xy – 2y2) 6x3 + 8y2 + 5xy + 4xy – 2y2 Associative Property 6x + 6y + 9xy 3 2 Combine like terms.

Example 1C: Adding Polynomials Horizontally (3x2y – 5x) + (4x + 7) + 6x2y (3x2y – 5x) + (4x + 7) + 6x2y 3x2y – 5x + 4x + 7 + 6x2y Associative Property 9x2y – x + 7 Combine like terms.

Example 2A Add. (3y4 + y2 + 6) + (5y4 + 2y2) (3y + y + 6) + (5y + 2y ) Associative Property 8y + 3y + 6 4 2 Combine like terms.

Example 2B Add. (9x3 + 6p2 + 3xy) + (8xy – 3p2) Associative Property 9x3+ 3p2 + 11xy Combine like terms.

Example 2C Add. (3z2w – 5x) + (2x + 8) + 6z2w (3z2w – 5x) + (2x + 8) + 6z2w 3z2w – 5x + 2x + 8 + 6z2w Associative Property 9z2w – 3x + 8 Combine like terms.

You can also add polynomials in a vertical format 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.

Example 3: Adding Polynomials Vertically A. (4x2 + 2x + 11) + (2x2 + 6x + 9) 4x2 + 2x + 11 + 2x2 + 6x + 9 Place like terms in columns. 6x2 + 8x + 20 Combine like terms.

Example 3: Adding Polynomials Vertically B. (3mn2 – 6m + 6n) + (5mn2 + 2m – n) C. (–x2y2 + 5x2) + (–2y2 + 2) + (x2 + 8) 3mn2 – 6m + 6n + 5mn2 + 2m – n Place like terms in columns. 8mn2 – 4m + 5n Combine like terms. –x2y2 + 5x2 –2y2 + 2 Place like terms in columns. + x2 + 8 –x2y2 + 6x2 – 2y2 + 10 Combine like terms.

Example 4 Add. A. (6x2 + 6x + 13) + (3x2+ 2x + 4) 6x2 + 6x + 13 + 3x2 + 2x + 4 Place like terms in columns. 9x2 + 8x + 17 Combine like terms.

Example 4 Add. B. (4mn2 + 6m + 2n) + (2mn2 – 2m – 2n) C. (x2y2 – 5x2) + (2y2 – 2) + (x2) 4mn2 + 6m + 2n + 2mn2 – 2m – 2n Place like terms in columns. 6mn2 + 4m Combine like terms. x2y2 – 5x2 2y2 – 2 Place like terms in columns. + x2 x2y2 – 4x2 + 2y2 – 2 Combine like terms.

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

Example 1: Finding the Opposite of a Polynomial Find the opposite of each polynomial. A. 8x3y4z2 –(8x3y4z2) –8x3y4z2 Distributive Property. B. –3x4 + 8x2 –(–3x4 + 8x2) 3x4 – 8x2 Distributive Property.

Additional Example 1: Finding the Opposite of a Polynomial Find the opposite of the polynomial. C. 9a6b4 + a4b2 – 1 –(9a6b4 + a4b2 – 1) –9a6b4 – a4b2 + 1 Distributive Property.

To subtract a polynomial, add its opposite.

Example 1: Subtracting Polynomials Horizontally A. (5x2 + 2x – 3) – (3x2 + 8x – 4) Add the opposite. = (5x2 + 2x – 3) + (–3x2 – 8x + 4) Associative property. = 5x2 + 2x – 3 – 3x2 – 8x + 4 = 2x2 – 6x + 1 Combine like terms.

Example 1: Subtracting Polynomials Horizontally B. (b2 + 4b – 1) – (7b2 – b – 1) = (b2 + 4b – 1) + (–7b2 + b + 1) Add the opposite. Associative property. = b2 + 4b – 1 – 7b2 + b + 1 = –6b2 + 5b Combine like terms.

Example 2A Subtract. (2y3 + 3y + 5) – (4y3 + 3y + 5) Add the opposite. = (2y3 + 3y + 5) + (–4y3 – 3y – 5) Associative property. = 2y3 + 3y + 5 – 4y3 – 3y – 5 = –2y3 Combine like terms.

Example 2B Subtract. (c3 + 2c2 + 3) – (4c3 – c2 – 1) = (c3 + 2c2 + 3) + (–4c3 + c2 + 1) Add the opposite. = c3 + 2c2 + 3 – 4c3 + c2 + 1 Associative property. = –3c3 + 3c2 + 4 Combine like terms.

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

Example 3: Subtracting Polynomials Vertically (2n2 – 4n + 9) – (6n2 – 7n + 5) (2n2 – 4n + 9) 2n2 – 4n + 9 – (6n2 – 7n + 5) + –6n2 + 7n – 5 Add the opposite. –4n2 + 3n + 4

Example 4: Subtracting Polynomials Vertically (10x2 + 2x – 7) – (x2 + 5x + 1) (10x2 + 2x – 7) 10x2 + 2x – 7 – (x2 + 5x + 1) + –x2 – 5x – 1 Add the opposite. 9x2 – 3x – 8

Example 5: Subtracting Polynomials Vertically (6a4 – 3a2 – 8) – (–2a4 + 7) (6a4 – 3a2 – 8) 6a4 – 3a2 – 8 – (–2a4 + 7) + 2a4 – 7 Rearrange as needed. 8a4 – 3a2 – 15

Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

Lesson Quiz: Part I Add. 1. (2m2 – 3m + 7) + (7m2 – 1) 2. (yz2 + 5yz + 7) + (2yz2 – yz) 3. 9m2 – 3m + 6 3yz2+ 4yz + 7 (2xy2 + 2x – 6) + (5xy2 + 3y + 8) 7xy + 2x + 3y + 2 2

Lesson Quiz Find the opposite of each polynomial. Subtract. 3. (3z2 – 7z + 6) – (2z2 + z – 12) 1. 3a2b2c3 –3a2b2c3 2. –3m3 + 2m2n 3m3 – 2m2n z2 – 8z + 18 4. –18h3 – (4h3 + h2 – 12h + 2) 5. (3b2c + 5bc2 – 8b2) – (4b2c + 2bc2 – c2) – 22h3 – h2 + 12h – 2 – b2c + 3bc2 – 8b2 + c2

Lesson Quiz for Student Response Systems 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 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 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 3. Subtract. (11p2 – 5p + 9) – (3p2 + p – 17) A. 14p2 + 4p – 8 B. 14p2 – 6p + 26 C. 8p2 – 6p + 26 D. 8p2 + 4p – 8