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Add and subtract polynomials.

VocabularyA monomial is a number, a variable, or the product of a number and one or more variables with whole number exponents.

The degree of a monomial is the sum of the exponents of the variables in the monomial.

A polynomial is a monomial or a sum of monomials, each called a term of the polynomial.

The degree of a polynomial is the greatest degree of its terms.

When a polynomial is written so that the exponents of a variable decrease from left to right, the coefficient of the first term is called the leading coefficient.

A polynomial with two terms is called a binomial.

A polynomial with three terms is called a trinomial.

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Study guideFor use with the lesson “Add and Subtract Polynomials”

Rewrite a polynomial

Write 12x 3 2 15x 1 13x 5 so that the exponents decrease from left to right. Identify the degree and the leading coefficient of the polynomial.

Solution

Consider the degree of each of the polynomial’s terms.

Degree is 3. Degree is 1. Degree is 5.

12x3 2 15x 1 13x5

The polynomial can be rewritten as 13x5 1 12x3 2 15x. The greatest degree is 5, so the degree of the polynomial is 5, and the leading coefficient is 13.

Exercises for Example 1

Write the polynomial so that the exponents decrease from left to right. Identify the degree and the leading coefficient of the polynomial.

1. 9 2 2x2 2. 16 1 3y3 1 2y 3. 6z3 1 7z2 2 3z5

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Subtract polynomials

Find the difference.

a. (3x2 2 9x) 2 (2x2 2 5x 1 6) b. (11x2 1 6x 2 1) 2 (2x2 2 7x 1 5)

Solution

a. Vertical format: Align like terms in vertical columns.

3x2 2 9x 3x2 2 9x

2 (2x2 2 5x 1 6) 2 2x2 1 5x 2 6

_______________ _____________

x2 2 4x 2 6

b. Horizontal format: Group like terms and simplify.

(11x2 1 6x 2 1) 2 (2x2 2 7x 1 5) 5 11x2 1 6x 2 1 2 2x2 1 7x 2 5

5 (11x2 2 2x2) 1 (6x 1 7x) 1 (21 2 5)

5 9x2 1 13x 2 6

Exercises for Examples 2 and 3

Find the sum or difference.

4. (2a2 1 7) 1 (7a2 1 4a 2 3)

5. (9b2 2 b 1 8) 1 (4b2 2 b 2 3)

6. (7c3 2 6c 1 4) 2 (9c3 2 5c2 2 c)

7. (d2 2 15d 1 10) 2 (212d2 1 8d 2 1)

ExAmplE 3

Add polynomials

Find the sum.

a. (3x4 2 2x3 1 5x2) 1 (7x2 1 9x32 2x) b. (7x22 3x 1 6) 1 (9x2 1 6x2 11)

Solution

a. Vertical format: Align like terms in vertical columns.

3x4 2 2x3 1 5x2

1 9x3 1 7x2 2 2x ______________________

3x4 1 7x3 1 12x2

2 2x

b. Horizontal format: Group like terms and simplify.

(7x2 2 3x 1 6) 1 (9x2 1 6x 2 11) 5 (7x2 1 9x2) 1 (23x 1 6x) 1 (6 2 11)

5 16x2 1 3x 2 5

ExAmplE 2

Study guide continuedFor use with the lesson “Add and Subtract Polynomials”

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8-11Algebra 1

Chapter Resource Book

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Lesson Add and Subtract Polynomials

Teaching Guide

1–2. Check students’ tiles. 3. 3x2 1 x 1 1 4. Because subtraction is the same as adding the opposite, find the opposite of each term in the second polynomial. Then use algebra tiles to represent each polynomial and proceed as you do when adding polynomials.

Investigating Algebra Activity

1. 3x2 1 5x 2 3; 1 1

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2. 22x 2 1; 2

2 2

3. 2x2 1 3x 2 2; 1

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4. 22x2 1 3x 1 7; 2 2

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5. To subtract polynomials, add the opposite. So, multiply each term in the subtracted polynomial by 21 and add. Then use algebra tiles to model the addition of the polynomials. 6. 2x 2 3 7. 2x2 2 8 8. 5x2 1 6 9. 23x2 1 3x 1 3

Practice Level A

1. 8n6; degree: 6; leading coefficient: 8 2. 29z 1 1; degree: 1; leading coefficient: 29 3. 2x5 1 4; degree: 5; leading coefficient: 2 4. 2x2 1 18x 1 2; degree: 2; leading coefficient: 21 5. 3y3 1 4y2 1 8; degree: 3; leading coefficient: 3 6. 220m3 1 m 1 5; degree: 3; leading coefficient: 220 7. 23a7 1 10a4 2 8; degree: 7; leading coefficient: 23 8. 6z4 1 z3 2 5z2 1 4z; degree: 4; leading coefficient: 6 9. h7 2 6h4 1 8h3; degree: 7; leading coefficient: 1 10. polynomial; degree: 2; monomial 11. not a polynomial; variable exponent 12. not a polynomial; negative exponent 13. polynomial; degree: 2; binomial 14. polynomial; degree: 2; trinomial 15. polynomial: degree: 3; binomial 16. 7x 1 9 17. 7m2 2 7 18. 9y2 1 5y 2 4 19. 2x2 1 3 20. 7a2 1 2a 2 6 21. 2m2 2 8m 1 3 22. 4x 1 4

23. 4x 1 9 24. B 5 0.014t2 1 0.13t 1 12 25. Area: 4x2 2 12πx 1 6π

Practice Level B

1. 4n5; degree: 5; leading coefficient: 4

2. 22x2 1 4x 1 3; degree: 2; leading coefficient: 22 3. 4y4 1 6y3 2 2y2 2 5; degree: 4; leading coefficient: 4 4. not a polynomial; variable exponent

5. polynomial; degree: 3; trinomial 6. not a polynomial; negative exponent 7. 5z2 1 3z 2 7 8. 5c2 2 3c 1 6

9. 3x2 1 6 10. 6b2 2 8b 1 1

11. 24m2 1 2m 2 3 12. 22m2 1 9m 2 1

13. 10x 1 2 14. 9x 2 1

15. Area: 17

} 4 x2 1 8x 2 32

16. P 5 1 }

6 t2 1 2t 1 200

Practice Level C

1. polynomial; degree: 0; monomial 2. not a polynomial; negative exponent 3. polynomial; degree: 2; trinomial

4. 3m3 1 4m2 2 m 1 2 5. 25y2 2 2y 1 9

6. c3 1 c2 2 9c 1 5 7. 24z2 1 4z 1 14

8. 14x4 2 3x3 2 7x2 2 3

9. 2x4 2 2x3 1 6x2 2 5x

10. f (x) 1 g(x) 5 6x3 2 3x2 1 2x 2 6; f (x) 2 g(x) 5 26x3 2 7x2 1 2x 1 4

11. 24a3b2 1 15a2b2 2 10a2b 1 5

12. 3m2n 2 11mn2 2 8n 1 2m

13. a. T 5 4.93t4 2 56.78t3 1 177.65t2 2 126.42t 1 1367.51 b. In 1997, 1367.51 thousand metric tons were produced and in 2003, 1129.19 thousand metric tons were produced. So more peat and perlite were produced in 1997.

14. a. N 5 187,443 1 13,857t; M 5 151,629 1 5457t b. 1997: $35,814; 2003: $86,214; Northeast: $83,142; Midwest: $32,742

Study Guide

1. 22x2 1 9; degree: 2; coefficient: 22

2. 3y3 1 2y 1 16; degree: 3; coefficient: 3

3. 23z5 1 6z3 1 7z2; degree: 5; coefficient: 23

4. 9a2 1 4a 1 4 5. 13b2 2 2b 1 5

6. 22c3 1 5c2 2 5c 1 4 7. 13d2 2 23d 1 11

Problem Solving Workshop: Worked Out Example

1. 22,055,300 people 2. $1,115,940

Answers for Chapter Polynomials and Factoring

Algebra 1Chapter Resource BookA12

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