7-1 Introduction to Polynomials
Name
Date
Determine whether each is a monomial.
13x3 ⫺ 9x
x
No, the expression
contains more than
one term.
No, the variable is
under a radical sign.
29x3y4z6
dⴚ1
Yes, the product of a
number and three
variables.
No, the variable has
a negative exponent.
Remember: A monomial is an algebraic expression that is a product of a constant and
a number of variables, each raised to a nonnegative power.
Are the following monomials like terms?
19x4 and 23x3
3x2y5 and ⫺29x2y5
⫺27d 5 and 27d 5
⫺18, c 2, and 12c 3
No, the exponents of
the variables are not
identical.
Yes, only their
numerical
coefficients differ.
Yes, only their
numerical
coefficients differ.
No, exponents of the
variables are not the
same. ⫺18 is a
constant.
Classify each polynomial and state its degree.
g2 ⫺ 4g ⫹ 29
The degree of g2 is 2.
The degree of 4g is 1.
The degree of 29 is 0.
29 ⫽ 29g0
There are 3 terms, so the polynomial is a
trinomial. The greatest degree of the terms is 2,
so the polynomial is quadratic.
x3 ⫹ 2x2
The degree of x3 is 3.
The degree of 2x2 is 2.
There are 2 terms, so the polynomial is a
binomial. The greatest degree of the terms is 3,
so the polynomial is cubic.
Determine whether the expression is a monomial. Explain. Check student’s responses.
Copyright © by William H. Sadlier, Inc. All rights reserved.
1. 2x ⫹ 7y
2. 9a ⫺ 7b ⫹ 3
No, the expression contains
more than one term.
5. 6aⴚ3
No
Yes
7. 2.3
g
6. 11yzⴚ8
No
4. ⫺9x2y2
3. 7xy2
8. 6.8
a
No
No
Yes
No
Tell whether the monomials are like monomials. If so, combine the like terms.
9. 3a and ⫺9a
Yes; 3 ⴙ (ⴚ9) ⴝ ⴚ6
ⴚ6a
13. 6d 2e and 3d 2e
Yes; 6 ⴙ 3 ⴝ 9; 9d2e
17. 3xyz, 7xz, and 8yz
No
10. 11ab and ⫺23ab
11. xy and xy2
Yes; 11 ⴙ (ⴚ23) ⴝ ⴚ12
ⴚ12ab
14. ⫺8fg 2 and 11fg 2
No
15. 13t and ⫺24v
Yes; ⴚ8 ⴙ 11 ⴝ 3; 3fg2
18. 2ac, ⫺5cb, and 2ab
No
No
19. 4b3, ⫺9b3, and 15b3
Yes; 4 ⴙ (ⴚ9) ⴙ 15 ⴝ 10
10b3
Lesson 7-1, pages 176–177.
12. x2y and xy
No
16. 91x and 91y
No
20. ⫺6w4, 2w4, and 24w4
Yes; ⴚ6 ⴙ 2 ⴙ 24 ⴝ 20
20w4
Chapter 7
169
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Simplify.
21. ⫺2a2b ⫹ 9ab2 ⫺ 4a2b ⫹ 3ab
22. 5m3n ⫺ 5mn2 ⫹ m3n ⫹ mn2
ⴚ6a2b ⴙ 9ab2 ⴙ 3ab
23. 15x2y ⫹ x2y ⫺ 2x2y ⫹ 2xy
14x2y ⴙ 2xy
6m3n ⴚ 4mn2
24. 3a2b ⫺ a2b2 ⫺ 2a2b ⫺ 3a2b2 ⫹ 5a2b
25. 4x2y3 ⫹ 2xy2 ⫺ x2y3 ⫹ 3x2y3 ⫹ 5xy2
6x2y3 ⴙ 7xy2
6a2b ⴚ 4a2b2
Classify each expression as a monomial, binomial, or trinomial and then state its degree.
26. 9k 4ᐉ2
27. 17m5n11
4ⴙ2ⴝ6
monomial; degree of 6
30. 7s4t 2 ⫺ 12t 9 ⫹ 8
monomial; degree of 16
31. 5m3n7 ⫹ 4m2 ⫺ 11
trinomial; degree of 9
29. ⫺8
28. 5
trinomial; degree of 10
monomial; degree of 0;
constant
32. 4a9 ⫺ 2a6b8
monomial; degree of 0;
constant
33. ⫺7j4k2 ⫺ 13k8 ⫹ 8j 4k2
binomial; degree of 14
trinomial; degree of 8
Write each polynomial in standard form. (Hint: Watch for like terms.)
2x ⴝ
and 11 ⴝ
3⬎2⬎1⬎0
ⴚ9x3 ⴙ 3x2 ⴙ 2x ⴙ 11
2x1
11x0
37. 11z5 ⫹ 3z2 ⫺ 11z5 ⫹ 31
3z2 ⴙ 31
35. 8x2 ⫺ 11x3 ⫹ 13x ⫹ 1
ⴚ11x3 ⴙ 8x2 ⴙ 13x ⴙ 1
38. 2.3m2 ⫺ 5.1m ⫺ 1.8 ⫹ 6.3m3
ⴚ4z ⴙ 24
39. 4.8n ⫺ 7.4n3 ⫺ 2.9n ⫹ 5.2n2
6.3m3 ⴙ 2.3m2 ⴚ 5.1m ⴚ 1.8
40. Show four different monomials of degree
3 that use at most two variables, x and y,
all with a coefficient of 1.
x3; x2y; xy2; y3
36. 9z4 ⫺ 9z4 ⫺ 4z ⫹ 24
ⴚ7.4n3 ⴙ 5.2n2 ⴙ 1.9n
41. A trinomial of degree 2 has terms with degree
2, 1 and 0. Give at least two different trinomials
that could fit this description.
Possible answer: x2 ⴙ x ⴙ 2; xy ⴙ x ⴙ 1
42. Write the polynomial below in standard form. Explain your reasoning.
4y4 ⫹ 2x4 ⫹ 6x3y ⫹ 6xy3 ⫺ 9x2y2
2x4 ⴙ 6x3y ⴚ 9x2y2 ⴙ 6xy3 ⴙ 4y4; arrange terms in descending order of powers of x
because x comes before y in the alphabet.
170
Chapter 7
Copyright © by William H. Sadlier, Inc. All rights reserved.
34. 3x2 ⫹ 11 ⫹ 2x ⫺ 9x3