lesson study guide 8 - woodbridge.k12.nj.us · exercises for example 1 write the polynomial so that...
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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
ExAmplE 1
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8-10Algebra 1Chapter Resource Book
CS10_CC_A1_MECR710730_C8L01SG.indd 10 5/14/11 12:18:42 AM
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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”
Less
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Lesson
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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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CS10_CC_A1_MECR710730_C8AK.indd 12 5/21/11 2:40:42 AM