4.1E: Exercises
- Page ID
- 108349
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Practice Makes Perfect
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
1. \(4p+11p\)
- Answer
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\(15p\)
2. \(−8y^3−5y^3\)
- Answer
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\(−13y^3\)
3. \((4a^2+9a−11)+(6a^2−5a+10)\)
- Answer
-
\(10a^2+4a-1\)
4. \((8m^2+12m−5)−(2m^2−7m−1)\)
- Answer
-
\(6m^2+19m−4\)
5. \((y^2−3y+12)+(5y^2−9)\)
- Answer
-
\(6y^2-3y+3\)
6. \((5u^2+8u)−(4u−7)\)
- Answer
-
\(5u^2+4u+7\)
7. Find the sum of \(8q^3−27\) and \(q^2+6q−2\).
- Answer
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\(2x^2−2x+23\)
8. Find the difference of \(x^2+6x+8\) and \(x^2−8x+15\).
- Answer
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\(14x-7\)
In the following exercises, simplify.
9. \(17mn^2−(−9mn^2)+3mn^2\)
- Answer
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\(29mn^2\)
10. \(18a−7b−21a\)
- Answer
-
\(−7b−3a\)
11. \(2pq^2−5p−3q^2\)
- Answer
-
\(2pq^2−5p−3q^2\)
12. \((6a^2+7)+(2a^2−5a−9)\)
- Answer
-
\(8a^2−5a−2\)
13. \((3p^2−4p−9)+(5p^2+14)\)
- Answer
-
\(8p^2-4p+5\)
14. \((7m^2−2m−5)−(4m^2+m−8)\)
- Answer
-
\(−3m-3m+3\)
15. \((7b^2−4b+3)−(8b^2−5b−7)\)
- Answer
-
\(-b^2-b-10\)
16. Subtract \((8y^2−y+9)\) from \( (11y^2−9y−5) \)
- Answer
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\(3y^2−8y−14\)
17. Find the difference of \((z^2−4z−12)\) and \((3z^2+2z−11)\)
- Answer
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\(-2z^2+6z+1\)
18. \((x^3−x^2y)−(4xy^2−y^3)+(3x^2y−xy^2)\)
- Answer
-
\(y^3-5xy^2+2x^2y+x^3\)
19. \((x^3−2x^2y)−(xy^2−3y^3)−(x^2y−4xy^2)\)
- Answer
-
\(3y^3+3xy^2-3x^2y+x^3\)
Multiply Monomials
In the following exercises, multiply the monomials.
20. \((−6p^4)(9p)\)
- Answer
-
\(-54p^{5}\)
21. \(\left(\frac{1}{3}c^2\right)(30c^8)\)
- Answer
-
\(10c^{10}\)
22. \((8x^2y^5)(7xy^6)\)
- Answer
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\(56x^3y^{11}\)
23. \( \left(\frac{2}{3}m^3n^6\right)\left(\frac{1}{6}m^4n^4\right)\)
- Answer
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\(\dfrac{m^7n^{10}}{9}\)
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
24. \(7(10−x)\)
- Answer
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\(70-7x\)
25. \(a^2(a^2−9a−36)\)
- Answer
-
\(a^4−9a^3−36a^2\)
26. \(−5y(125y^3−1)\)
- Answer
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\(-625y^4+5y\)
27. \((4n−5)(2n^3)\)
- Answer
-
\(8n^4−10n^3\)
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using any method:
28. \((a+5)(a+2)\)
- Answer
-
\(a^2+7a+10\)
29. \((y−4)(y+12)\)
- Answer
-
\(y^2+8y−48\)
30. \((3x+1)(2x−7)\)
- Answer
-
\(6x^2-19x-7\)
31. \((6p−11)(3p−10)\)
- Answer
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\(18p^2−93p+110\)
32. \((n+8)(n+1)\)
- Answer
-
\(n^2+9n+8\)
33. \((k+6)(k−9)\)
- Answer
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\(k^2−3k−54\)
34. \((5u−3)(u+8)\)
- Answer
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\(5u^2+37u-24\)
35. \((2y−9)(5y−7)\)
- Answer
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\(10y^2−59y+63\)
36. \((p+4)(p+7)\)
- Answer
-
\(p^2+11p+28\)
37. \((x−8)(x+9)\)
- Answer
-
\(x^2+x−72\)
38. \((3c+1)(9c−4)\)
- Answer
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\(27c^2-3c-4\)
39. \((10a−1)(3a−3)\)
- Answer
-
\(30a^2−33a+3\)
Multiply a Polynomial by a Polynomial
In the following exercises, multiply using any method
40. \((x+1)(x^2−3x−21)\)
- Answer
-
\(x^3-2x^2-24x-21\)
41. \((5b−2)(3b^2+b−9)\)
- Answer
-
\(15b^3−b^2−47b+18\)
42. \((m+6)(m^2−7m−30)\)
- Answer
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\(m^3-m^2-72m-180\)
43. \((4y−1)(6y^2−12y+5)\)
- Answer
-
\(24y^2−54y^2+32y−5\)
Multiply Special Products
In the following exercises, square each binomial using the Binomial Squares Pattern.
44. \((x−3)^2\)
- Answer
-
\(x^2 - 6x + 9\)
45. \((x−5)^2\)
- Answer
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\(x^2 - 10x + 25\)
46. \((x+1)^2\)
- Answer
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\(x^2 + 2x + 1\)
47. \((x+3)^2\)
- Answer
-
\(x^2 + 6x + 9\)
48. \((2x−y)^2\)
- Answer
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\(4x^2-4xy+y^2\)
49. \((x+\dfrac{3}{4})^2\)
- Answer
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\(x^2+\dfrac{3}{2}x+\dfrac{9}{16}\)
50. \((8p^3−3)^2\)
- Answer
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\(64p^6-48p^3+9\)
51. \((5p+7q)^2\)
- Answer
-
\(25p^2+70pq+49q^2\)
In the following exercises, multiply each pair of conjugates using the Product of Conjugates.
52. \((y+5)(y−5)\)
- Answer
-
\(y^2−25\)
53. \((x-4)(x+4)\)
- Answer
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\(x^2−16\)
54. \((a+3)(a−3)\)
- Answer
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\(a^2−9\)
55. \((m+1)(m-1)\)
- Answer
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\(m^2−1\)
56. \((x+9)(x−9)\)
- Answer
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\(x^2-81\)
57. \((3y+5)(3y−5)\)
- Answer
-
\(9y^2−25\)
58. \((6x+y)(6x−y)\)
- Answer
-
\(36x^2−y^2\)
59. \((a+\dfrac{2}{3}b)(a−\dfrac{2}{3}b)\)
- Answer
-
\(a^2−\dfrac{4}{9}b^2\)
60. \((12x^3−7y^2)(12x^3+7y^2)\)
- Answer
-
\(144x^6−49y^4\)
61. \((13a^2−8b4)(13a^2+8b^4)\)
- Answer
-
\(169a^4-64b^8\)