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LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product’s selling price in order to maximize the company’s profit. A landscape architect may be interested in finding a formula for the area of a patio deck. To find these quantities, you need to be able to add, subtract, multiply, and divide polynomials.
Explain
• Degree of a Polynomial
CONCEPT 1: ADDING AND SUBTRACTING POLYNOMIALS
• Writing Terms in Descending Order
Definitions
Concept 1 has sections on • Definitions
• Evaluating a Polynomial • Adding Polynomials • Subtracting Polynomials
A monomial is an algebraic expression that contains exactly one term. The term may be a constant, or the product of a constant and one or more variables. The exponent of any variable must be a nonnegative integer (that is, a whole number). The following are monomials: 12
x2
5wy3
1 gt 2 2
4.35T
A monomial in one variable, x, can be written in the form axr, where a is any real number and r is a nonnegative integer.
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
383
The following are not monomials: 2 3 The denominator contains a variable with a positive exponent. x r
So the term cannot be written in the form ax where r is a nonnegative integer.
x2 3
There is a squared variable under a cube root symbol. So the term cannot be written in the form axr where r is a nonnegative integer.
A polynomial is the sum of one or more monomials. Here are some examples: 2x3 5x 2
x 2
3xy2 7x 5y 1
5
A polynomial with one, two, or three terms has a special name. Number of terms
Name A polynomial with 4 terms is called a “four term polynomial.” A polynomial with 5 terms is called a “five term polynomial,” and so on.
Examples
monomial
1
x, 5y, 3xy3, 5
binomial
2
x 1, 2x2 3, 5xy3 4x3y2
trinomial
3
x2 2x 1, 3x2y3 xy 5
Example 6.2.1
Determine if each expression is a polynomial. 24 b. 3x 2
a. 4w 3
x
c. x3 2
d. x 2
Solution a. The expression is a polynomial. It has one term, so it is a monomial. The term has the form aw r, where a 4 and r 3. b. The expression is not a polynomial. 24 x
cannot be written in the form axr where r is a The term 2 nonnegative integer. Remember: x 0 1, for x 0 x1 x
c. The expression is not a polynomial. The term x3 cannot be written in the form axr where r is a nonnegative integer. d. The expression is a polynomial. It has two terms, so it is a binomial. Each term can be written in the form axr: x2 1x 1 2x0
384
TOPIC 6 EXPONENTS AND POLYNOMIALS
Degree of a Polynomial The degree of a term of a polynomial is the sum of the exponents of the variables in that term. 6x3y2 xy2 35x4.
For example, consider this trinomial: The degree of the first term is 5.
6x3y2 degree 3 2 5
The degree of the second term is 3.
xy2 x1y2 degree 1 2 3
The degree of the last term is 4. In 35, the exponent does not contribute to the degree because the base, 3, is not a variable.
35x4 degree 4
The degree of a polynomial is equal to the degree of the term with the highest degree. In this polynomial, the term with the highest degree has degree 5. So this polynomial has degree 5.
degree 5 degree 3 degree 4 6x 3y2 x1y2 35x4 The polynomial has degree 5.
Writing Terms in Descending Order The terms of a polynomial in one variable are usually arranged by degree, in descending order, when read from left to right. For example, this polynomial contains one variable, x. The terms of the polynomial are arranged by degree in descending order.
x3 7x2 4x 2 x3 7x2 4x1 2x0 degree degree 3 2
degree 1
degree 0
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
385
Example 6.2.2
Arrange the terms of this polynomial in descending order and determine the degree of the polynomial: 7x3 8 2x x 4 Solution Write 8 as 8x0. Write 2x as 2x1.
7x3 8x0 2x1 x4
Arrange the terms by degree in descending order (4, 3, 1, 0).
x4 7x3 2x1 8x0
The last two terms may be written without exponents.
x4 7x3 2x 8
The term of highest degree is x4. The degree of x4 is 4. So, the degree of this polynomial is 4.
Evaluating a Polynomial To evaluate a polynomial, we replace each variable with the given number, then simplify. Example 6.2.3
Evaluate this polynomial when w 3 and y 2: 6w 2 4wy y 4 5 Solution Substitute 3 for w and 2 for y.
6(3)2 4(3)(2) (2)4 5
First, do the calculations with the exponents.
6(9)
4(3)(2) 16 5
Multiply.
54
24
Add and subtract.
19
16 5
Adding Polynomials To add polynomials, combine like terms. Recall that like terms are terms that have the same variables raised to the same power. That is, like terms have the same variables with the same exponents. Like terms 3x, 12x 8xy2, 5.6xy2 24, 11 4xy, 6yx NOT like terms 7x, 5xy 3x2y3,
386
2x3y2
TOPIC 6 EXPONENTS AND POLYNOMIALS
The variables do not match. The powers of x do not match. The powers of y do not match.
Example 6.2.4
Find: (5x3 13x2 7) (16x3 8x2 x 15) Solution
(5x3 13x2 7) (16x3 8x2 x 15)
Remove the parentheses.
5x3 13x2 7 16x3 8x2 x 15
Write like terms next to each other.
5x3 16x3 13x2 8x2 x 7 15
Combine like terms.
21x3 5x2 x 8
We can also place one polynomial beneath the other and add like terms. 5x3 13x2 7 16x3 8x2 x 15 21x3 5x2 x 8
Example 6.2.5
Find the sum of (3z3 2zy2 6y3) and (15z3 5zy2 4z2 ). Solution Write the sum.
(3z3 2zy2 6y3) (15z3 5zy2 4z2 )
Remove the parentheses.
3z3 2zy2 6y3 15z3 5zy2 4z2
Write like terms next to each other.
Combine like terms.
18z3 3zy2 6y3 4z2
3z3
15z3
2zy2
5zy2
6y3
4z2
We can also place one polynomial beneath the other and add like terms. 3z3 2zy2 6y 3 15z3 5zy2 4z2 3 2 3 18z 3zy 6y 4z2
Subtracting Polynomials To subtract one polynomial from another, add the first polynomial to the opposite of the polynomial being subtracted. To find the opposite of a polynomial, multiply each term by 1.
Here’s a way to find the opposite of a polynomial: Change the sign of each term.
For example: The opposite of 5x2 is 5x2. The opposite of 2x 7 is 2x 7. Example 6.2.6
Find: (18w2 w 32) (40 13w2) Solution
(18w2 w 32) (40 13w2)
Change the subtraction to addition of the opposite.
(18w2 w 32) (1)(40 13w2)
Remove the parentheses.
18w2 w 32 40 13w2
Write like terms next to each other.
18w2 13w2 w 32 40
Combine like terms.
5w2 w 72
So, (18w2 w 32) (40 13w2) 5w2 w 72.
We can also place one polynomial beneath the other and subtract like terms. 18w2 w 32 (13w2 40) To do the subtraction, we change the sign of each term being subtracted, then add. 18w2 w 32 (13w2 w 40) 5w2 w 72
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
387
Example 6.2.7
Subtract (15z2 5yz2 4y3) from (6y3 10z3 2yz2). Solution We can also place one polynomial beneath the other and subtract like terms. 6y 3 10z 3 2yz 2 (4y 3 5yz 2 15z 2) To do the subtraction, we change the sign of each term being subtracted, then add. 6y 3 10z 3 02yz 2 (4y 3 05yz 2 15z 2) 2y 3 10z 3 10yz 2 15z 2
Be careful! “Subtract A from B” means B A. The order is important. Write the difference.
(6y3 10z3 2yz2) (15z2 5yz2 4y3)
Change the subtraction to addition of the opposite. (6y3 10z3 2yz2) (1)(15z2 5yz2 4y3) Remove the parentheses. 6y3 10z3 2yz2 15z2 5yz2 4y3 Write like terms next to each other. 6y3 4y3 10z3 2yz2 5yz2 15z2 Combine like terms.
2y3 10z3 7yz2 15z2
Here is a summary of this concept from Interactive Mathematics.
388
TOPIC 6 EXPONENTS AND POLYNOMIALS
CONCEPT 2: MULTIPLYING AND DIVIDING POLYNOMIALS Multiplying a Monomial By a Monomial To find the product of two monomials, multiply the coefficients. Then, use the Multiplication Property of Exponents to combine variable factors that have the same base.
Concept 2 has sections on • Multiplying a Monomial by a Monomial • Multiplying a Polynomial by a Monomial • Dividing a Monomial by a Monomial • Dividing a Polynomial by a Monomial
Example 6.2.8
Find: 7m3n4 6mn2 Solution Write the coefficients next to each other. Write the factors with base m next to each other, and write the factors with base n next to each other.
7m3n4 6mn2
Use the Multiplication Property of Exponents.
(7 6)(m3 1n4 2)
Simplify.
42m4n6
(7 6)(m3 m1)(n4 n2) Multiplication Property of Exponents: xm xn xm n
Example 6.2.9 1 3
Find: w3x7y 6w2y5 Solution 1 3 1 6 (w3 w2)(x7)(y1 y5) 3
Write the coefficients next to each other. Write the factors with base w next to each other, and write the factors with base y next to each other.
w3x7y 6w2y5
Use the Multiplication Property of Exponents.
6 (w3 2)(x7)(y1 5)
Simplify.
2w5x7y6
13
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
389
Example 6.2.10
Find: (5x3y)(3x5)(2xy5) Solution Write the coefficients next to each other. Write the factors with base x next to each other and write the factors with base y next to each other.
(5x3y)(3x5)(2xy5)
Use the Multiplication Property of Exponents.
(5 3 2)(x3 5 1)(y1 5)
Simplify.
30x9y6
(5 3 2)(x3 x5 x1)(y1 y5)
Multiplying a Polynomial By a Monomial To multiply a monomial by a polynomial with more than one term, use the Distributive Property to distribute the monomial to each term in the polynomial. Example 6.2.11
Find: 8w3y(4w2y5 w4) Solution
8w3y(4w2y5 w4)
Multiply each term in the polynomial by the monomial, 8w3y.
(8w3y)(4w2y5) (8w3y)(w4)
Within each term, write the coefficients next to each other. Write the factors with base w next to each other and write the factors with base y next to each other. (8 4)(w3 w2)(y y5) (8)(w3 w4)(y)
390
Use the Multiplication Property of Exponents.
(8 4)(w3 2y1 5) (8)(w3 4y)
Simplify.
TOPIC 6 EXPONENTS AND POLYNOMIALS
32w5y6
8w7y
Example 6.2.12
Find: 5x4(3x2y2 2xy2 x3y) Solution
5x4(3x2y2 2xy2 x3y)
Multiply each term in the polynomial by the monomial, 5x4. (5x4)(3x2y2) (5x4)(2xy2) (5x 4)(x 3y) Within each term, write the coefficients next to each other. Write the factors with base x next to each other and write the factors with base y next to each other. (5 3)(x4x2y2) (5 2)(x4x1y2) (5 1)(x4x3y) Use the Multiplication Property of Exponents. (5 3)(x4 2y2) (5 2)(x4 1y2) (5 1)(x4 3y) Simplify.
15x6y2
10x5y2
5x7y
Dividing a Monomial By a Monomial To divide a monomial by a monomial, use the Division Property of Exponents. (Assume that any variable in the denominator is not equal to zero.) Division Property of Exponents xm xm n for m n and x 0 xn xm 1 for m n and x 0 xn xn m
Example 6.2.13
Find: 36w5xy3 9w2y7 Solution
36w5xy3 9w2y7
Rewrite the problem using a division bar.
2 7
Cancel the common factor, 9, in the numerator and denominator.
2 7
Use the Division Property of Exponents.
73
Simplify.
36w5xy3 9w y 4w5xy3 wy
4w5 2x y 4w3x y4
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
391
Dividing a Polynomial By a Monomial a b ab , where c 0 c c c
When you added fractions, you learned: If we exchange the expressions on either side of the equals sign, we have:
ab a b c c c
We will use this property to divide a polynomial by a monomial. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Example 6.2.14
Find: (27w5x3y2 12w3x2y) 3w2xy Solution
(27w5x3y2 12w3x2y) 3w2xy
Rewrite the problem using a division bar.
2
Divide each term of the polynomial by the monomial.
2 2
Cancel the common factor, 3, in each fraction.
2 2
Use the Division Property of Exponents.
Note that y 1 1 y 0 1.
9w3x2y 4wx
27w5x3y2 12w3x2y 3w xy
12w3x2y 3w xy
27w5x3y2 3w xy
4w3x2y w xy
9w5x3y2 w xy
9w5 2x3 1y2 1 1
4w3 2x2 1y1 1 1
Example 6.2.15
A landscape architect is designing a patio. She wants to estimate the cost of the patio for various widths and lengths. a. Construct an expression for the area of the patio in terms of x and y. b. If the brick she will use costs $4.50 per square foot, find the cost of the brick for a patio that is 10 feet wide by 40 feet long. x 2y y y
y x 2y length = x
392
TOPIC 6 EXPONENTS AND POLYNOMIALS
width = y y
Solution a. The patio is made up of two triangles and a rectangle. Recall two formulas from geometry: Area of a rectangle length width
Area lw
1 2
1 2
Area of a triangle (base)(height)
Area bh
Express the area of each triangle in terms of y.
Area bh
1 2
1 2
(y)(y)
Each triangle has base y and height y.
1 2
y2 Area lw
Express the area of the rectangle in terms of x and y. The rectangle has length (x 2y) and width y. The area of the patio is the sum of the areas of the two triangles and the rectangle.
(x 2y)y xy 2y 2 area of
area of
area of
Area triangle rectangle triangle 1 2
Substitute the expressions for area.
y2
Simplify.
xy y2
1 2
xy 2y2 y2
Therefore, the area of the patio in terms of x and y is xy y2. b. The length of the base of the patio is x. This is 40 feet. The width of the patio, y, is 10 feet. 20 feet 10 feet 20 feet
10 feet
width = 10 feet
10 feet 10 feet
length = 40 feet
In the formula for the area of the patio, substitute 40 feet for x and 10 feet for y.
Area xy y 2 (40 feet)(10 feet) (10 feet)2 300 feet 2
The cost of the patio is the price per square foot times the number of square feet.
Cost 2 300 feet 24
$4.50 1 foot 4
$1350
The bricks for the patio will cost $1350.
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
393
Here is a summary of this concept from Interactive Mathematics.
394
TOPIC 6 EXPONENTS AND POLYNOMIALS
Checklist Lesson 6.2 Here is what you should know after completing this lesson.
Words and Phrases degree of a term degree of a polynomial evaluate a polynomial
monomial polynomial binomial trinomial
Ideas and Procedures ❶ Definition of a Polynomial Determine whether a given expression is a polynomial.
Example 6.2.1b 24 Determine if 3x is a polynomial. 2 x
See also: Example 6.2.1a, c, d Apply 1-4 ❷ Degree of a Polynomial Arrange the terms of a polynomial in descending order by degree and determine the degree of the polynomial.
Example 6.2.2 Arrange the terms of this polynomial in descending order and determine the degree of the polynomial: 7x3 8 2x x 4 See also: Apply 5-7
❸ Evaluate a Polynomial Evaluate a polynomial when given a specific value for each variable.
Example 6.2.3 Evaluate this polynomial when w 3 and y 2: 6w2 4wy y4 5 See also: Apply 8-13
❹ Add Polynomials Find the sum of polynomials.
Example 6.2.5 Find the sum of (3z3 2zy2 6y3) and (15z3 5zy2 4z2). See also: Example 6.2.4 Apply 14-22
❺ Subtract Polynomials Find the difference of polynomials.
Example 6.2.7 Subtract (15z2 5yz2 4y3) from (6y3 10z3 2yz2). See also: Example 6.2.6 Apply 23-28
LESSON 6.2 POLYNOMIAL OPERATIONS I
CHECKLIST
395
❻ Multiply Monomials Find the product of monomials.
Example 6.2.10 Find: (5x3y)(3x5)(2xy5) See also: Example 6.2.8, 6.2.9 Apply 29-35
❼ Multiply a Monomial by a Polynomial Find the product of a monomial and a polynomial.
Example 6.2.12 Find: 5x4(3x2y2 2xy2 x3y) See also: Example 6.2.11 Apply 36-41
❽ Divide a Monomial by a Monomial Find the quotient of two monomials.
Example 6.2.13 Find: 36w5xy3 9w2y7 See also: Apply 42-50
❾ Divide a Polynomial by a Monomial Find the quotient of a polynomial divided by a monomial.
Example 6.2.14 Find: (27w5x3y2 12w3x2y) 3w2xy See also: Example 6.2.15 Apply 51-56
396
TOPIC 6 EXPONENTS AND POLYNOMIALS
Homework Homework Problems Circle the homework problems assigned to you by the computer, then complete them below. 9. Angelina works at a pet store. Today, she is cleaning three fish tanks. These polynomials describe the volumes of the tanks:
Explain Adding and Subtracting Polynomials
Tank 1: xy2 Tank 2: x2y 2y3 4xy2 3
1. Circle the algebraic expression that is a polynomial.
Tank 3: x2y 5xy2 6y3
1 4 1 3y3 3y2 5 4 1 3 3y2 5 4y
Write a polynomial that describes the total volume of the three tanks. Hint: Add the polynomials.
2 3y3 3y 5
volume ________ 10. Angelina has three fish tanks to clean. These polynomials describe their volumes.
2. Write m beside the monomial, b beside the binomial, and t beside the trinomial.
Tank 1: xy2
____ 34x x2 z ____
wxy3z2
____
pn2
Tank 2: x2y 2y3 4xy2 3 Tank 3: x2y 5xy2 6y3
13n3
3. Given the polynomial 3y
2y3
4y5
What is the total volume of the fish tanks if x 3 feet and y 1.5 feet?
2:
volume ________ cubic feet
a. write the terms in descending order.
11. Find: (w2yz 3w3 2wyz2 4wyz) (4wy2z 3w2yz 2wyz2) (2wyz 3)
b. find the degree of each term. c. find the degree of the polynomial. 4. Find: (3w 12w3 2) (15w 2w3 4w5 3) 5. Find: (2v3 6v2 2) (5v v3 4v7 3) 6.
1 Evaluate xy 3y2 5x3 when x 2 and y 4. 4
7. Find: (s2t s3t3 4st2 27) (3st2 2st 8s3t3 13t 36)
12. Find: (tu2v 4t2u2v 9t3uv 3tv) (3t2u2 2tv t3) (4t2u2v 3tv 2tu2v) (6t3uv 2tv)
Multiplying and Dividing Polynomials 13. Find: xyz x2y2z2 14. Find: 3p2r 2p3qr
8. Find: (12x3y 9x2y2 6xy y 7) (7xy x y 11x3y 3x2y2 4)
1 2
15. Find: 6t3u2v11 tu2v4 16. Find: 3y(2x3 3x2y) 17. Find: 5p2r3(2pr p2r2) 18. Find: t3uv4(2tu 3uv 4tv 5) 19. Write 12w7x3y2z6 4w2x2y3z6 as a fraction and simplify.
LESSON 6.2 POLYNOMIAL OPERATIONS I
HOMEWORK
397
20. Write (36x3y3 15x2y5) 9x2y as a fraction and simplify. 21. Find:
15a7b4d2
10a4b9c3d
22. Tony is an algebra student. This is how he answered a question on a test: (2t8u3 4t4u9 6t12u6) 2t4u3 t2u 2tu3 3t3u2 Is his answer right or wrong? Why? Circle the most appropriate response. The answer is right. The answer is wrong. Tony divided the exponents rather than adding them. The correct answer is t12u6 t8u12 t16u9. The answer is wrong. The terms need to be ordered by degree. The correct answer is 3t3u2 t2u 2tu3. The answer is wrong. Tony divided the exponents rather than subtracting them. The correct answer is t4 2u6 3t8u3. The answer is wrong. Tony shouldn't have canceled the numerical coefficients. The correct answer is 2t2u 4tu3 6t3u2.
398
TOPIC 6 EXPONENTS AND POLYNOMIALS
23. Find: (16x2y4 20x3y5) 12xy2 24. Find: (20t5u11 5t3u5 30tu6v5) 10t4u5
Apply Practice Problems Here are some additional practice problems for you to try.
Adding and Subtracting Polynomials
5. Find the degree of the polynomial 8a3b5 11a2b3 7b6.
1. Circle the algebraic expressions below that are polynomials.
6. Find the degree of the polynomial 12m4n7 16m12.
2xy 5xz 2 6x 3x
7. Find the degree of the polynomial 7x3y2z 3x2y3z4 6z7.
9y 2 13yz 8z 2
8. Evaluate 2x2 8x 11 when x 1.
24x 5
9. Evaluate x3 3x2 x 1 when x 2.
15a3
5a8
2. Circle the algebraic expressions below that are polynomials. 3 8xy y
17x 3 3w 7wz 1
10. Evaluate 2x2 5x 8 when x 3. 11. Evaluate x2y xy2 when x 2 and y 3. 12. Evaluate 5mn 4mn2 8m n when m 4 and n 2. 13. Evaluate 3uv 6u2v 2u v 4 when u 2 and v 4.
7x2 13x 8y2
14. Find: (3x2 7x) (x2 5)
12x2 3x3
15. Find: (5x2 4x 8) (x2 7x)
3. Identify each polynomial below as a monomial, a binomial, or a trinomial. a. 17x 24z b. 13ab2 5 c. m n 10 d. 42a2b4c e. 73 65x 21y 4. Identify each polynomial below as a monomial, a binomial, or a trinomial. a. 25 6xyz 4x b. 2xyz3 c. x y 1
16. Find: (6a2 8a 10) (3a2 2a 7) 17. Find: (12m2n3 7m2n2 14mn) (3m2n3 5m2n2 7mn) 18. Find: (10x4y3 9x2y3 6xy2 x) (28x4y3 14x2y3 3xy2 x) 19. Find: (13a3b2 6a2b 5ab3 b) (2a3b2 2a2b 4ab3 b) 20. Find: (11u5v4w3 6u3v2w) (6u5v4w3 11u3v2w) 21. Find: (7xy2z3 19x2yz2 26x3y3z) (13xy2z3 11x2yz2 16x3y3z)
d. 36 3xyz
22. Find: (9a4b2c 3a2b3c 5abc) (2abc 6a4b2c 2) (3a2b3c 5)
e. 32x2y
23. Find: (5x3 7x) (x3 8) 24. Find: (9a2 7ab 14b) (3a2 7b)
LESSON 6.2 POLYNOMIAL OPERATIONS I
APPLY
399
25. Find: (2y2 6xy 3y) (y2 y) 26. Find: (8x3 9x2 17) (5x3 3x2 15) 27. Find: (9a5b3 8a4b 6b) (2a5b3 12a4b 3b) 28. Find: (7x4y2 3x2y 5x) (9x4y2 3x2y 2x)
Multiplying and Dividing Polynomials 29. Find:
3y4
5y
43. 44. 45. 46. 47.
30. Find: 5x3 2x 31. Find: 5a5 9a4
48.
32. Find: 3x3 12x4
49.
33. Find: 4x3y5 7xy3
50.
34. Find: 7a5b6c3 8ab3c 35. Find: 3w2x3y2z 2x2yz2
51.
36. Find: 4y3(3y2 5y 10)
52.
37. Find: 2a3b2(3a4b5 5ab3 6a)
53.
38. Find:
2xy3(2x6
5x4
y2)
39. Find: 5a2b2(4a2 2a2b 7ab2 3b)
54.
40. Find: 4mn3(3m2n 12mn2 6m 7n2)
55.
41. Find: 4x3y3(3x3 7xy2 2xy y)
56.
400
TOPIC 6 EXPONENTS AND POLYNOMIALS
9x3y 3x 20a5b6 Find: 4a3b 12x4y6 Find: 3x2y 32a7b9c Find: 5 12a b6c2 15m6n10 Find: 10n4p3 24x6y2z7 Find: 16wx3z2 27a4b3c12d Find: 15ac7d3 42mn6p3q4 Find: 28m2nq5 36w2x3y7z Find: 5 21w y2z2 32a3 24a5 Find: 8a2 21m2 18mn3 Find: 3mn 14x 8x4y2 Find: 2xy 24a2b2c3 4ab4c5 Find: 16abc3 32x2y3z4 8x5yz7 Find: 3 16x y3z4 32r4st2 3r2st5 Find: 12r3s2t
42. Find: 2
Evaluate Practice Test Take this practice test to be sure that you are prepared for the final quiz in Evaluate. 1. Circle the expressions that are polynomials. 2 p3r 3p2q 2r 5 5 3 c15 c11 3
7 14 2 x2 3xy y2 3x
325 t2 s 5 m5n4o3p2r
2. Write m beside the monomial(s), b beside the binomial(s), and t beside the trinomial(s). a. ___ w5x4 b. ___
2x2
3. Given the polynomial 3w3 13w2 7w5 8w8 2, write the terms in descending order by degree. 4. Find: a. (5x3y 8x2y2 3xy y3 13) (2xy 6 y2 4y3 2x3y) b. (5x3y 8x2y2 3xy y3 13) (2xy 6 y2 4y3 2x3y) 5. Find: x3y2w x5yw4
36
1 2 1 c. ___ x17 x12 3 3 3
d. ___ 27 e. ___ 27x3 2x2y3
6. Find: n2p3(3n 2n3p2 35p4) 7. Find: 21x5y2z7 14xyz 8. Find: (15t3u2v 5t5uv2) 10tuv2
2 3
f. ___ x2 3xy y2
LESSON 6.2 POLYNOMIAL OPERATIONS I
EVALUATE
401
Explain
• Degree of a Polynomial
CONCEPT 1: ADDING AND SUBTRACTING POLYNOMIALS
• Writing Terms in Descending Order
Definitions
Concept 1 has sections on • Definitions
• Evaluating a Polynomial • Adding Polynomials • Subtracting Polynomials
A monomial is an algebraic expression that contains exactly one term. The term may be a constant, or the product of a constant and one or more variables. The exponent of any variable must be a nonnegative integer (that is, a whole number). The following are monomials: 12
x2
5wy3
1 gt 2 2
4.35T
A monomial in one variable, x, can be written in the form axr, where a is any real number and r is a nonnegative integer.
LESSON 6.2 POLYNOMIAL OPERATIONS I
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383
The following are not monomials: 2 3 The denominator contains a variable with a positive exponent. x r
So the term cannot be written in the form ax where r is a nonnegative integer.
x2 3
There is a squared variable under a cube root symbol. So the term cannot be written in the form axr where r is a nonnegative integer.
A polynomial is the sum of one or more monomials. Here are some examples: 2x3 5x 2
x 2
3xy2 7x 5y 1
5
A polynomial with one, two, or three terms has a special name. Number of terms
Name A polynomial with 4 terms is called a “four term polynomial.” A polynomial with 5 terms is called a “five term polynomial,” and so on.
Examples
monomial
1
x, 5y, 3xy3, 5
binomial
2
x 1, 2x2 3, 5xy3 4x3y2
trinomial
3
x2 2x 1, 3x2y3 xy 5
Example 6.2.1
Determine if each expression is a polynomial. 24 b. 3x 2
a. 4w 3
x
c. x3 2
d. x 2
Solution a. The expression is a polynomial. It has one term, so it is a monomial. The term has the form aw r, where a 4 and r 3. b. The expression is not a polynomial. 24 x
cannot be written in the form axr where r is a The term 2 nonnegative integer. Remember: x 0 1, for x 0 x1 x
c. The expression is not a polynomial. The term x3 cannot be written in the form axr where r is a nonnegative integer. d. The expression is a polynomial. It has two terms, so it is a binomial. Each term can be written in the form axr: x2 1x 1 2x0
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TOPIC 6 EXPONENTS AND POLYNOMIALS
Degree of a Polynomial The degree of a term of a polynomial is the sum of the exponents of the variables in that term. 6x3y2 xy2 35x4.
For example, consider this trinomial: The degree of the first term is 5.
6x3y2 degree 3 2 5
The degree of the second term is 3.
xy2 x1y2 degree 1 2 3
The degree of the last term is 4. In 35, the exponent does not contribute to the degree because the base, 3, is not a variable.
35x4 degree 4
The degree of a polynomial is equal to the degree of the term with the highest degree. In this polynomial, the term with the highest degree has degree 5. So this polynomial has degree 5.
degree 5 degree 3 degree 4 6x 3y2 x1y2 35x4 The polynomial has degree 5.
Writing Terms in Descending Order The terms of a polynomial in one variable are usually arranged by degree, in descending order, when read from left to right. For example, this polynomial contains one variable, x. The terms of the polynomial are arranged by degree in descending order.
x3 7x2 4x 2 x3 7x2 4x1 2x0 degree degree 3 2
degree 1
degree 0
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
385
Example 6.2.2
Arrange the terms of this polynomial in descending order and determine the degree of the polynomial: 7x3 8 2x x 4 Solution Write 8 as 8x0. Write 2x as 2x1.
7x3 8x0 2x1 x4
Arrange the terms by degree in descending order (4, 3, 1, 0).
x4 7x3 2x1 8x0
The last two terms may be written without exponents.
x4 7x3 2x 8
The term of highest degree is x4. The degree of x4 is 4. So, the degree of this polynomial is 4.
Evaluating a Polynomial To evaluate a polynomial, we replace each variable with the given number, then simplify. Example 6.2.3
Evaluate this polynomial when w 3 and y 2: 6w 2 4wy y 4 5 Solution Substitute 3 for w and 2 for y.
6(3)2 4(3)(2) (2)4 5
First, do the calculations with the exponents.
6(9)
4(3)(2) 16 5
Multiply.
54
24
Add and subtract.
19
16 5
Adding Polynomials To add polynomials, combine like terms. Recall that like terms are terms that have the same variables raised to the same power. That is, like terms have the same variables with the same exponents. Like terms 3x, 12x 8xy2, 5.6xy2 24, 11 4xy, 6yx NOT like terms 7x, 5xy 3x2y3,
386
2x3y2
TOPIC 6 EXPONENTS AND POLYNOMIALS
The variables do not match. The powers of x do not match. The powers of y do not match.
Example 6.2.4
Find: (5x3 13x2 7) (16x3 8x2 x 15) Solution
(5x3 13x2 7) (16x3 8x2 x 15)
Remove the parentheses.
5x3 13x2 7 16x3 8x2 x 15
Write like terms next to each other.
5x3 16x3 13x2 8x2 x 7 15
Combine like terms.
21x3 5x2 x 8
We can also place one polynomial beneath the other and add like terms. 5x3 13x2 7 16x3 8x2 x 15 21x3 5x2 x 8
Example 6.2.5
Find the sum of (3z3 2zy2 6y3) and (15z3 5zy2 4z2 ). Solution Write the sum.
(3z3 2zy2 6y3) (15z3 5zy2 4z2 )
Remove the parentheses.
3z3 2zy2 6y3 15z3 5zy2 4z2
Write like terms next to each other.
Combine like terms.
18z3 3zy2 6y3 4z2
3z3
15z3
2zy2
5zy2
6y3
4z2
We can also place one polynomial beneath the other and add like terms. 3z3 2zy2 6y 3 15z3 5zy2 4z2 3 2 3 18z 3zy 6y 4z2
Subtracting Polynomials To subtract one polynomial from another, add the first polynomial to the opposite of the polynomial being subtracted. To find the opposite of a polynomial, multiply each term by 1.
Here’s a way to find the opposite of a polynomial: Change the sign of each term.
For example: The opposite of 5x2 is 5x2. The opposite of 2x 7 is 2x 7. Example 6.2.6
Find: (18w2 w 32) (40 13w2) Solution
(18w2 w 32) (40 13w2)
Change the subtraction to addition of the opposite.
(18w2 w 32) (1)(40 13w2)
Remove the parentheses.
18w2 w 32 40 13w2
Write like terms next to each other.
18w2 13w2 w 32 40
Combine like terms.
5w2 w 72
So, (18w2 w 32) (40 13w2) 5w2 w 72.
We can also place one polynomial beneath the other and subtract like terms. 18w2 w 32 (13w2 40) To do the subtraction, we change the sign of each term being subtracted, then add. 18w2 w 32 (13w2 w 40) 5w2 w 72
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
387
Example 6.2.7
Subtract (15z2 5yz2 4y3) from (6y3 10z3 2yz2). Solution We can also place one polynomial beneath the other and subtract like terms. 6y 3 10z 3 2yz 2 (4y 3 5yz 2 15z 2) To do the subtraction, we change the sign of each term being subtracted, then add. 6y 3 10z 3 02yz 2 (4y 3 05yz 2 15z 2) 2y 3 10z 3 10yz 2 15z 2
Be careful! “Subtract A from B” means B A. The order is important. Write the difference.
(6y3 10z3 2yz2) (15z2 5yz2 4y3)
Change the subtraction to addition of the opposite. (6y3 10z3 2yz2) (1)(15z2 5yz2 4y3) Remove the parentheses. 6y3 10z3 2yz2 15z2 5yz2 4y3 Write like terms next to each other. 6y3 4y3 10z3 2yz2 5yz2 15z2 Combine like terms.
2y3 10z3 7yz2 15z2
Here is a summary of this concept from Interactive Mathematics.
388
TOPIC 6 EXPONENTS AND POLYNOMIALS
CONCEPT 2: MULTIPLYING AND DIVIDING POLYNOMIALS Multiplying a Monomial By a Monomial To find the product of two monomials, multiply the coefficients. Then, use the Multiplication Property of Exponents to combine variable factors that have the same base.
Concept 2 has sections on • Multiplying a Monomial by a Monomial • Multiplying a Polynomial by a Monomial • Dividing a Monomial by a Monomial • Dividing a Polynomial by a Monomial
Example 6.2.8
Find: 7m3n4 6mn2 Solution Write the coefficients next to each other. Write the factors with base m next to each other, and write the factors with base n next to each other.
7m3n4 6mn2
Use the Multiplication Property of Exponents.
(7 6)(m3 1n4 2)
Simplify.
42m4n6
(7 6)(m3 m1)(n4 n2) Multiplication Property of Exponents: xm xn xm n
Example 6.2.9 1 3
Find: w3x7y 6w2y5 Solution 1 3 1 6 (w3 w2)(x7)(y1 y5) 3
Write the coefficients next to each other. Write the factors with base w next to each other, and write the factors with base y next to each other.
w3x7y 6w2y5
Use the Multiplication Property of Exponents.
6 (w3 2)(x7)(y1 5)
Simplify.
2w5x7y6
13
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
389
Example 6.2.10
Find: (5x3y)(3x5)(2xy5) Solution Write the coefficients next to each other. Write the factors with base x next to each other and write the factors with base y next to each other.
(5x3y)(3x5)(2xy5)
Use the Multiplication Property of Exponents.
(5 3 2)(x3 5 1)(y1 5)
Simplify.
30x9y6
(5 3 2)(x3 x5 x1)(y1 y5)
Multiplying a Polynomial By a Monomial To multiply a monomial by a polynomial with more than one term, use the Distributive Property to distribute the monomial to each term in the polynomial. Example 6.2.11
Find: 8w3y(4w2y5 w4) Solution
8w3y(4w2y5 w4)
Multiply each term in the polynomial by the monomial, 8w3y.
(8w3y)(4w2y5) (8w3y)(w4)
Within each term, write the coefficients next to each other. Write the factors with base w next to each other and write the factors with base y next to each other. (8 4)(w3 w2)(y y5) (8)(w3 w4)(y)
390
Use the Multiplication Property of Exponents.
(8 4)(w3 2y1 5) (8)(w3 4y)
Simplify.
TOPIC 6 EXPONENTS AND POLYNOMIALS
32w5y6
8w7y
Example 6.2.12
Find: 5x4(3x2y2 2xy2 x3y) Solution
5x4(3x2y2 2xy2 x3y)
Multiply each term in the polynomial by the monomial, 5x4. (5x4)(3x2y2) (5x4)(2xy2) (5x 4)(x 3y) Within each term, write the coefficients next to each other. Write the factors with base x next to each other and write the factors with base y next to each other. (5 3)(x4x2y2) (5 2)(x4x1y2) (5 1)(x4x3y) Use the Multiplication Property of Exponents. (5 3)(x4 2y2) (5 2)(x4 1y2) (5 1)(x4 3y) Simplify.
15x6y2
10x5y2
5x7y
Dividing a Monomial By a Monomial To divide a monomial by a monomial, use the Division Property of Exponents. (Assume that any variable in the denominator is not equal to zero.) Division Property of Exponents xm xm n for m n and x 0 xn xm 1 for m n and x 0 xn xn m
Example 6.2.13
Find: 36w5xy3 9w2y7 Solution
36w5xy3 9w2y7
Rewrite the problem using a division bar.
2 7
Cancel the common factor, 9, in the numerator and denominator.
2 7
Use the Division Property of Exponents.
73
Simplify.
36w5xy3 9w y 4w5xy3 wy
4w5 2x y 4w3x y4
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
391
Dividing a Polynomial By a Monomial a b ab , where c 0 c c c
When you added fractions, you learned: If we exchange the expressions on either side of the equals sign, we have:
ab a b c c c
We will use this property to divide a polynomial by a monomial. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Example 6.2.14
Find: (27w5x3y2 12w3x2y) 3w2xy Solution
(27w5x3y2 12w3x2y) 3w2xy
Rewrite the problem using a division bar.
2
Divide each term of the polynomial by the monomial.
2 2
Cancel the common factor, 3, in each fraction.
2 2
Use the Division Property of Exponents.
Note that y 1 1 y 0 1.
9w3x2y 4wx
27w5x3y2 12w3x2y 3w xy
12w3x2y 3w xy
27w5x3y2 3w xy
4w3x2y w xy
9w5x3y2 w xy
9w5 2x3 1y2 1 1
4w3 2x2 1y1 1 1
Example 6.2.15
A landscape architect is designing a patio. She wants to estimate the cost of the patio for various widths and lengths. a. Construct an expression for the area of the patio in terms of x and y. b. If the brick she will use costs $4.50 per square foot, find the cost of the brick for a patio that is 10 feet wide by 40 feet long. x 2y y y
y x 2y length = x
392
TOPIC 6 EXPONENTS AND POLYNOMIALS
width = y y
Solution a. The patio is made up of two triangles and a rectangle. Recall two formulas from geometry: Area of a rectangle length width
Area lw
1 2
1 2
Area of a triangle (base)(height)
Area bh
Express the area of each triangle in terms of y.
Area bh
1 2
1 2
(y)(y)
Each triangle has base y and height y.
1 2
y2 Area lw
Express the area of the rectangle in terms of x and y. The rectangle has length (x 2y) and width y. The area of the patio is the sum of the areas of the two triangles and the rectangle.
(x 2y)y xy 2y 2 area of
area of
area of
Area triangle rectangle triangle 1 2
Substitute the expressions for area.
y2
Simplify.
xy y2
1 2
xy 2y2 y2
Therefore, the area of the patio in terms of x and y is xy y2. b. The length of the base of the patio is x. This is 40 feet. The width of the patio, y, is 10 feet. 20 feet 10 feet 20 feet
10 feet
width = 10 feet
10 feet 10 feet
length = 40 feet
In the formula for the area of the patio, substitute 40 feet for x and 10 feet for y.
Area xy y 2 (40 feet)(10 feet) (10 feet)2 300 feet 2
The cost of the patio is the price per square foot times the number of square feet.
Cost 2 300 feet 24
$4.50 1 foot 4
$1350
The bricks for the patio will cost $1350.
LESSON 6.2 POLYNOMIAL OPERATIONS I
EXPLAIN
393
Here is a summary of this concept from Interactive Mathematics.
394
TOPIC 6 EXPONENTS AND POLYNOMIALS
Checklist Lesson 6.2 Here is what you should know after completing this lesson.
Words and Phrases degree of a term degree of a polynomial evaluate a polynomial
monomial polynomial binomial trinomial
Ideas and Procedures ❶ Definition of a Polynomial Determine whether a given expression is a polynomial.
Example 6.2.1b 24 Determine if 3x is a polynomial. 2 x
See also: Example 6.2.1a, c, d Apply 1-4 ❷ Degree of a Polynomial Arrange the terms of a polynomial in descending order by degree and determine the degree of the polynomial.
Example 6.2.2 Arrange the terms of this polynomial in descending order and determine the degree of the polynomial: 7x3 8 2x x 4 See also: Apply 5-7
❸ Evaluate a Polynomial Evaluate a polynomial when given a specific value for each variable.
Example 6.2.3 Evaluate this polynomial when w 3 and y 2: 6w2 4wy y4 5 See also: Apply 8-13
❹ Add Polynomials Find the sum of polynomials.
Example 6.2.5 Find the sum of (3z3 2zy2 6y3) and (15z3 5zy2 4z2). See also: Example 6.2.4 Apply 14-22
❺ Subtract Polynomials Find the difference of polynomials.
Example 6.2.7 Subtract (15z2 5yz2 4y3) from (6y3 10z3 2yz2). See also: Example 6.2.6 Apply 23-28
LESSON 6.2 POLYNOMIAL OPERATIONS I
CHECKLIST
395
❻ Multiply Monomials Find the product of monomials.
Example 6.2.10 Find: (5x3y)(3x5)(2xy5) See also: Example 6.2.8, 6.2.9 Apply 29-35
❼ Multiply a Monomial by a Polynomial Find the product of a monomial and a polynomial.
Example 6.2.12 Find: 5x4(3x2y2 2xy2 x3y) See also: Example 6.2.11 Apply 36-41
❽ Divide a Monomial by a Monomial Find the quotient of two monomials.
Example 6.2.13 Find: 36w5xy3 9w2y7 See also: Apply 42-50
❾ Divide a Polynomial by a Monomial Find the quotient of a polynomial divided by a monomial.
Example 6.2.14 Find: (27w5x3y2 12w3x2y) 3w2xy See also: Example 6.2.15 Apply 51-56
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TOPIC 6 EXPONENTS AND POLYNOMIALS
Homework Homework Problems Circle the homework problems assigned to you by the computer, then complete them below. 9. Angelina works at a pet store. Today, she is cleaning three fish tanks. These polynomials describe the volumes of the tanks:
Explain Adding and Subtracting Polynomials
Tank 1: xy2 Tank 2: x2y 2y3 4xy2 3
1. Circle the algebraic expression that is a polynomial.
Tank 3: x2y 5xy2 6y3
1 4 1 3y3 3y2 5 4 1 3 3y2 5 4y
Write a polynomial that describes the total volume of the three tanks. Hint: Add the polynomials.
2 3y3 3y 5
volume ________ 10. Angelina has three fish tanks to clean. These polynomials describe their volumes.
2. Write m beside the monomial, b beside the binomial, and t beside the trinomial.
Tank 1: xy2
____ 34x x2 z ____
wxy3z2
____
pn2
Tank 2: x2y 2y3 4xy2 3 Tank 3: x2y 5xy2 6y3
13n3
3. Given the polynomial 3y
2y3
4y5
What is the total volume of the fish tanks if x 3 feet and y 1.5 feet?
2:
volume ________ cubic feet
a. write the terms in descending order.
11. Find: (w2yz 3w3 2wyz2 4wyz) (4wy2z 3w2yz 2wyz2) (2wyz 3)
b. find the degree of each term. c. find the degree of the polynomial. 4. Find: (3w 12w3 2) (15w 2w3 4w5 3) 5. Find: (2v3 6v2 2) (5v v3 4v7 3) 6.
1 Evaluate xy 3y2 5x3 when x 2 and y 4. 4
7. Find: (s2t s3t3 4st2 27) (3st2 2st 8s3t3 13t 36)
12. Find: (tu2v 4t2u2v 9t3uv 3tv) (3t2u2 2tv t3) (4t2u2v 3tv 2tu2v) (6t3uv 2tv)
Multiplying and Dividing Polynomials 13. Find: xyz x2y2z2 14. Find: 3p2r 2p3qr
8. Find: (12x3y 9x2y2 6xy y 7) (7xy x y 11x3y 3x2y2 4)
1 2
15. Find: 6t3u2v11 tu2v4 16. Find: 3y(2x3 3x2y) 17. Find: 5p2r3(2pr p2r2) 18. Find: t3uv4(2tu 3uv 4tv 5) 19. Write 12w7x3y2z6 4w2x2y3z6 as a fraction and simplify.
LESSON 6.2 POLYNOMIAL OPERATIONS I
HOMEWORK
397
20. Write (36x3y3 15x2y5) 9x2y as a fraction and simplify. 21. Find:
15a7b4d2
10a4b9c3d
22. Tony is an algebra student. This is how he answered a question on a test: (2t8u3 4t4u9 6t12u6) 2t4u3 t2u 2tu3 3t3u2 Is his answer right or wrong? Why? Circle the most appropriate response. The answer is right. The answer is wrong. Tony divided the exponents rather than adding them. The correct answer is t12u6 t8u12 t16u9. The answer is wrong. The terms need to be ordered by degree. The correct answer is 3t3u2 t2u 2tu3. The answer is wrong. Tony divided the exponents rather than subtracting them. The correct answer is t4 2u6 3t8u3. The answer is wrong. Tony shouldn't have canceled the numerical coefficients. The correct answer is 2t2u 4tu3 6t3u2.
398
TOPIC 6 EXPONENTS AND POLYNOMIALS
23. Find: (16x2y4 20x3y5) 12xy2 24. Find: (20t5u11 5t3u5 30tu6v5) 10t4u5
Apply Practice Problems Here are some additional practice problems for you to try.
Adding and Subtracting Polynomials
5. Find the degree of the polynomial 8a3b5 11a2b3 7b6.
1. Circle the algebraic expressions below that are polynomials.
6. Find the degree of the polynomial 12m4n7 16m12.
2xy 5xz 2 6x 3x
7. Find the degree of the polynomial 7x3y2z 3x2y3z4 6z7.
9y 2 13yz 8z 2
8. Evaluate 2x2 8x 11 when x 1.
24x 5
9. Evaluate x3 3x2 x 1 when x 2.
15a3
5a8
2. Circle the algebraic expressions below that are polynomials. 3 8xy y
17x 3 3w 7wz 1
10. Evaluate 2x2 5x 8 when x 3. 11. Evaluate x2y xy2 when x 2 and y 3. 12. Evaluate 5mn 4mn2 8m n when m 4 and n 2. 13. Evaluate 3uv 6u2v 2u v 4 when u 2 and v 4.
7x2 13x 8y2
14. Find: (3x2 7x) (x2 5)
12x2 3x3
15. Find: (5x2 4x 8) (x2 7x)
3. Identify each polynomial below as a monomial, a binomial, or a trinomial. a. 17x 24z b. 13ab2 5 c. m n 10 d. 42a2b4c e. 73 65x 21y 4. Identify each polynomial below as a monomial, a binomial, or a trinomial. a. 25 6xyz 4x b. 2xyz3 c. x y 1
16. Find: (6a2 8a 10) (3a2 2a 7) 17. Find: (12m2n3 7m2n2 14mn) (3m2n3 5m2n2 7mn) 18. Find: (10x4y3 9x2y3 6xy2 x) (28x4y3 14x2y3 3xy2 x) 19. Find: (13a3b2 6a2b 5ab3 b) (2a3b2 2a2b 4ab3 b) 20. Find: (11u5v4w3 6u3v2w) (6u5v4w3 11u3v2w) 21. Find: (7xy2z3 19x2yz2 26x3y3z) (13xy2z3 11x2yz2 16x3y3z)
d. 36 3xyz
22. Find: (9a4b2c 3a2b3c 5abc) (2abc 6a4b2c 2) (3a2b3c 5)
e. 32x2y
23. Find: (5x3 7x) (x3 8) 24. Find: (9a2 7ab 14b) (3a2 7b)
LESSON 6.2 POLYNOMIAL OPERATIONS I
APPLY
399
25. Find: (2y2 6xy 3y) (y2 y) 26. Find: (8x3 9x2 17) (5x3 3x2 15) 27. Find: (9a5b3 8a4b 6b) (2a5b3 12a4b 3b) 28. Find: (7x4y2 3x2y 5x) (9x4y2 3x2y 2x)
Multiplying and Dividing Polynomials 29. Find:
3y4
5y
43. 44. 45. 46. 47.
30. Find: 5x3 2x 31. Find: 5a5 9a4
48.
32. Find: 3x3 12x4
49.
33. Find: 4x3y5 7xy3
50.
34. Find: 7a5b6c3 8ab3c 35. Find: 3w2x3y2z 2x2yz2
51.
36. Find: 4y3(3y2 5y 10)
52.
37. Find: 2a3b2(3a4b5 5ab3 6a)
53.
38. Find:
2xy3(2x6
5x4
y2)
39. Find: 5a2b2(4a2 2a2b 7ab2 3b)
54.
40. Find: 4mn3(3m2n 12mn2 6m 7n2)
55.
41. Find: 4x3y3(3x3 7xy2 2xy y)
56.
400
TOPIC 6 EXPONENTS AND POLYNOMIALS
9x3y 3x 20a5b6 Find: 4a3b 12x4y6 Find: 3x2y 32a7b9c Find: 5 12a b6c2 15m6n10 Find: 10n4p3 24x6y2z7 Find: 16wx3z2 27a4b3c12d Find: 15ac7d3 42mn6p3q4 Find: 28m2nq5 36w2x3y7z Find: 5 21w y2z2 32a3 24a5 Find: 8a2 21m2 18mn3 Find: 3mn 14x 8x4y2 Find: 2xy 24a2b2c3 4ab4c5 Find: 16abc3 32x2y3z4 8x5yz7 Find: 3 16x y3z4 32r4st2 3r2st5 Find: 12r3s2t
42. Find: 2
Evaluate Practice Test Take this practice test to be sure that you are prepared for the final quiz in Evaluate. 1. Circle the expressions that are polynomials. 2 p3r 3p2q 2r 5 5 3 c15 c11 3
7 14 2 x2 3xy y2 3x
325 t2 s 5 m5n4o3p2r
2. Write m beside the monomial(s), b beside the binomial(s), and t beside the trinomial(s). a. ___ w5x4 b. ___
2x2
3. Given the polynomial 3w3 13w2 7w5 8w8 2, write the terms in descending order by degree. 4. Find: a. (5x3y 8x2y2 3xy y3 13) (2xy 6 y2 4y3 2x3y) b. (5x3y 8x2y2 3xy y3 13) (2xy 6 y2 4y3 2x3y) 5. Find: x3y2w x5yw4
36
1 2 1 c. ___ x17 x12 3 3 3
d. ___ 27 e. ___ 27x3 2x2y3
6. Find: n2p3(3n 2n3p2 35p4) 7. Find: 21x5y2z7 14xyz 8. Find: (15t3u2v 5t5uv2) 10tuv2
2 3
f. ___ x2 3xy y2
LESSON 6.2 POLYNOMIAL OPERATIONS I
EVALUATE
401
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