{"id":279,"date":"2026-01-30T23:00:16","date_gmt":"2026-01-30T23:00:16","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/polynomial-equations-get-stronger-key-precalculus-practice-page-answer-keys\/"},"modified":"2026-01-30T23:02:49","modified_gmt":"2026-01-30T23:02:49","slug":"polynomial-equations-get-stronger-key-precalculus-practice-page-answer-keys","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/polynomial-equations-get-stronger-key-precalculus-practice-page-answer-keys\/","title":{"raw":"Polynomial Equations: Get Stronger Key -- Precalculus Practice Page Answer Keys","rendered":"Polynomial Equations: Get Stronger Key — Precalculus Practice Page Answer Keys"},"content":{"raw":"\n\n\t\t
\n\t\t\t
\n\t
\n\t\t

<\/p>\n\t\t

Polynomial Equations: Get Stronger Key<\/h1>\n\t\t\t\t\t\t\t\t<\/div>\n\t
\n\t\t\t\t

Dividing Polynomials Solutions<\/h2>

1. The binomial is a factor of the polynomial.<\/p>

3. [latex]x+6+\\frac{5}{x - 1}\\text{,}\\text{quotient:}x+6\\text{,}\\text{remainder:}\\text{5}[\/latex]<\/p>

7. [latex]x - 5\\text{,}\\text{quotient: }x - 5\\text{,}\\text{remainder: }\\text{0}[\/latex]<\/p>

13. [latex]2{x}^{2}-3x+5\\text{,}\\text{quotient:}2{x}^{2}-3x+5\\text{,}\\text{remainder: }\\text{0}[\/latex]<\/p>

15. [latex]2{x}^{2}+2x+1+\\frac{10}{x - 4}[\/latex]<\/p>

19. [latex]3{x}^{2}-11x+34-\\frac{106}{x+3}[\/latex]<\/p>

23. [latex]4{x}^{2}-21x+84-\\frac{323}{x+4}[\/latex]<\/p>

29. [latex]{x}^{3}-3x+1[\/latex]<\/p>

43. [latex]\\text{Quotient: }4{x}^{2}+8x+16\\text{,}\\text{remainder: }-1[\/latex]<\/p>

45. [latex]\\text{Quotient: }3{x}^{2}+3x+5\\text{,}\\text{remainder: }0[\/latex]<\/p>

61. [latex]2x+3[\/latex]<\/p>

63. [latex]x+2[\/latex]<\/p>

Complex Numbers Solutions<\/h2>

1. Add the real parts together and the imaginary parts together.<\/p>

3. i<\/em> times i<\/em> equals \u20131, which is not imaginary. (answers vary)<\/p>

7. [latex]14+7i[\/latex]<\/p>

9. [latex]-\\frac{23}{29}+\\frac{15}{29}i[\/latex]<\/p>

13.
\"Graph<\/p>

15.
\"Graph<\/p>

17. [latex]8-i[\/latex]<\/p>

19. [latex]-11+4i[\/latex]<\/p>

23. [latex]6+15i[\/latex]<\/p>

25. [latex]-16+32i[\/latex]<\/p>

27. [latex]-4 - 7i[\/latex]<\/p>

29. 25<\/p>

33. [latex]4 - 6i[\/latex]<\/p>

35. [latex]\\frac{2}{5}+\\frac{11}{5}i[\/latex]<\/p>

37. 15i<\/em><\/p>

39. [latex]1+i\\sqrt{3}[\/latex]<\/p>

41. 1<\/p>

43. \u20131<\/p>

Zeros of Polynomial Functions Solutions<\/h2>

3. Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.<\/p>

5. Polynomial functions can have repeated zeros, so the fact that number is a zero doesn\u2019t preclude it being a zero again.<\/p>

7. \u2013106<\/p>

9. 0<\/p>

11. 255<\/p>

13. \u20131<\/p>

15. \u20132, 1, [latex]\\frac{1}{2}[\/latex]<\/p>

21. [latex]-\\frac{5}{2}, \\sqrt{6}, -\\sqrt{6}[\/latex]<\/p>

23. [latex]2, -4, -\\frac{3}{2}[\/latex]<\/p>

25. 4, \u20134, \u20135<\/p>

29. [latex]\\frac{1}{2}, \\frac{1+\\sqrt{5}}{2}, \\frac{1-\\sqrt{5}}{2}[\/latex]<\/p>

31. [latex]\\frac{3}{2}[\/latex]<\/p>

33. 2, 3, \u20131, \u20132<\/p>

37. [latex]-1, -1, \\sqrt{5}, -\\sqrt{5}[\/latex]<\/p>

41. [latex]2, 3+2i, 3 - 2i[\/latex]<\/p>

45. [latex]-\\frac{1}{2}, 1+4i, 1 - 4i[\/latex]<\/p>

57. [latex]\\pm 5, \\pm 1, \\pm \\frac{5}{2}[\/latex]<\/p>

59. [latex]\\pm 1, \\pm \\frac{1}{2}, \\pm \\frac{1}{3}, \\pm \\frac{1}{6}[\/latex]<\/p>

61. [latex]1, \\frac{1}{2}, -\\frac{1}{3}[\/latex]<\/p>

63. [latex]2, \\frac{1}{4}, -\\frac{3}{2}[\/latex]<\/p>

65. [latex]\\frac{5}{4}[\/latex]<\/p>

71. 8 by 4 by 6 inches<\/p>

75. 8 by 5 by 3 inches<\/p>

Inverses and Radical Functions Solutions<\/h2>

1. It can be too difficult or impossible to solve for x<\/em> in terms of y<\/em>.<\/p>

3. We will need a restriction on the domain of the answer.<\/p>

5. [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x}+4[\/latex]<\/p>

7. [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x+3}-1[\/latex]<\/p>

9. [latex]{f}^{-1}\\left(x\\right)=-\\sqrt{\\frac{x - 5}{3}}[\/latex]<\/p>

11. [latex]f\\left(x\\right)=\\sqrt{9-x}[\/latex]<\/p>

31. [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x+6}+3[\/latex]<\/p>

13. [latex]{f}^{-1}\\left(x\\right)=\\sqrt[3]{x - 5}[\/latex]<\/p>

17. [latex]{f}^{-1}\\left(x\\right)=\\frac{{x}^{2}-1}{2},\\left[0,\\infty \\right)[\/latex]<\/p>

21. [latex]{f}^{-1}\\left(x\\right)={\\left(\\frac{x - 9}{2}\\right)}^{3}[\/latex]<\/p>

23. [latex]{f}^{-1}\\left(x\\right)={\\frac{2 - 8x}{x}}[\/latex]<\/p>

25. [latex]{f}^{-1}\\left(x\\right)=\\frac{7x - 3}{1-x}[\/latex]<\/p>

35. [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x}+4[\/latex]
\"Graph<\/p>

37. [latex]{f}^{-1}\\left(x\\right)=\\sqrt[3]{1-x}[\/latex]
\"Graph<\/p>

41. [latex]{f}^{-1}\\left(x\\right)=\\sqrt{\\frac{1}{x}}[\/latex]
\"Graph<\/p>

57. [latex]t\\left(h\\right)=\\sqrt{\\frac{200-h}{4.9}}[\/latex], 5.53 seconds<\/p>

59. [latex]r\\left(V\\right)=\\sqrt[3]{\\frac{3V}{4\\pi }}[\/latex], 3.63 feet<\/p> \n\t<\/div>\n\t\t\t\n\t\t\t<\/div>\n\n\t\t<\/div>\n\t\n","rendered":"

\n
\n
\n

\n

Polynomial Equations: Get Stronger Key<\/h1>\n<\/p><\/div>\n
\n

Dividing Polynomials Solutions<\/h2>\n

1. The binomial is a factor of the polynomial.<\/p>\n

3. [latex]x+6+\\frac{5}{x - 1}\\text{,}\\text{quotient:}x+6\\text{,}\\text{remainder:}\\text{5}[\/latex]<\/p>\n

7. [latex]x - 5\\text{,}\\text{quotient: }x - 5\\text{,}\\text{remainder: }\\text{0}[\/latex]<\/p>\n

13. [latex]2{x}^{2}-3x+5\\text{,}\\text{quotient:}2{x}^{2}-3x+5\\text{,}\\text{remainder: }\\text{0}[\/latex]<\/p>\n

15. [latex]2{x}^{2}+2x+1+\\frac{10}{x - 4}[\/latex]<\/p>\n

19. [latex]3{x}^{2}-11x+34-\\frac{106}{x+3}[\/latex]<\/p>\n

23. [latex]4{x}^{2}-21x+84-\\frac{323}{x+4}[\/latex]<\/p>\n

29. [latex]{x}^{3}-3x+1[\/latex]<\/p>\n

43. [latex]\\text{Quotient: }4{x}^{2}+8x+16\\text{,}\\text{remainder: }-1[\/latex]<\/p>\n

45. [latex]\\text{Quotient: }3{x}^{2}+3x+5\\text{,}\\text{remainder: }0[\/latex]<\/p>\n

61. [latex]2x+3[\/latex]<\/p>\n

63. [latex]x+2[\/latex]<\/p>\n

Complex Numbers Solutions<\/h2>\n

1. Add the real parts together and the imaginary parts together.<\/p>\n

3. i<\/em> times i<\/em> equals \u20131, which is not imaginary. (answers vary)<\/p>\n

7. [latex]14+7i[\/latex]<\/p>\n

9. [latex]-\\frac{23}{29}+\\frac{15}{29}i[\/latex]<\/p>\n

13.
\"Graph<\/p>\n

15.
\"Graph<\/p>\n

17. [latex]8-i[\/latex]<\/p>\n

19. [latex]-11+4i[\/latex]<\/p>\n

23. [latex]6+15i[\/latex]<\/p>\n

25. [latex]-16+32i[\/latex]<\/p>\n

27. [latex]-4 - 7i[\/latex]<\/p>\n

29. 25<\/p>\n

33. [latex]4 - 6i[\/latex]<\/p>\n

35. [latex]\\frac{2}{5}+\\frac{11}{5}i[\/latex]<\/p>\n

37. 15i<\/em><\/p>\n

39. [latex]1+i\\sqrt{3}[\/latex]<\/p>\n

41. 1<\/p>\n

43. \u20131<\/p>\n

Zeros of Polynomial Functions Solutions<\/h2>\n

3. Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.<\/p>\n

5. Polynomial functions can have repeated zeros, so the fact that number is a zero doesn\u2019t preclude it being a zero again.<\/p>\n

7. \u2013106<\/p>\n

9. 0<\/p>\n

11. 255<\/p>\n

13. \u20131<\/p>\n

15. \u20132, 1, [latex]\\frac{1}{2}[\/latex]<\/p>\n

21. [latex]-\\frac{5}{2}, \\sqrt{6}, -\\sqrt{6}[\/latex]<\/p>\n

23. [latex]2, -4, -\\frac{3}{2}[\/latex]<\/p>\n

25. 4, \u20134, \u20135<\/p>\n

29. [latex]\\frac{1}{2}, \\frac{1+\\sqrt{5}}{2}, \\frac{1-\\sqrt{5}}{2}[\/latex]<\/p>\n

31. [latex]\\frac{3}{2}[\/latex]<\/p>\n

33. 2, 3, \u20131, \u20132<\/p>\n

37. [latex]-1, -1, \\sqrt{5}, -\\sqrt{5}[\/latex]<\/p>\n

41. [latex]2, 3+2i, 3 - 2i[\/latex]<\/p>\n

45. [latex]-\\frac{1}{2}, 1+4i, 1 - 4i[\/latex]<\/p>\n

57. [latex]\\pm 5, \\pm 1, \\pm \\frac{5}{2}[\/latex]<\/p>\n

59. [latex]\\pm 1, \\pm \\frac{1}{2}, \\pm \\frac{1}{3}, \\pm \\frac{1}{6}[\/latex]<\/p>\n

61. [latex]1, \\frac{1}{2}, -\\frac{1}{3}[\/latex]<\/p>\n

63. [latex]2, \\frac{1}{4}, -\\frac{3}{2}[\/latex]<\/p>\n

65. [latex]\\frac{5}{4}[\/latex]<\/p>\n

71. 8 by 4 by 6 inches<\/p>\n

75. 8 by 5 by 3 inches<\/p>\n

Inverses and Radical Functions Solutions<\/h2>\n

1. It can be too difficult or impossible to solve for x<\/em> in terms of y<\/em>.<\/p>\n

3. We will need a restriction on the domain of the answer.<\/p>\n

5. [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x}+4[\/latex]<\/p>\n

7. [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x+3}-1[\/latex]<\/p>\n

9. [latex]{f}^{-1}\\left(x\\right)=-\\sqrt{\\frac{x - 5}{3}}[\/latex]<\/p>\n

11. [latex]f\\left(x\\right)=\\sqrt{9-x}[\/latex]<\/p>\n

31. [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x+6}+3[\/latex]<\/p>\n

13. [latex]{f}^{-1}\\left(x\\right)=\\sqrt[3]{x - 5}[\/latex]<\/p>\n

17. [latex]{f}^{-1}\\left(x\\right)=\\frac{{x}^{2}-1}{2},\\left[0,\\infty \\right)[\/latex]<\/p>\n

21. [latex]{f}^{-1}\\left(x\\right)={\\left(\\frac{x - 9}{2}\\right)}^{3}[\/latex]<\/p>\n

23. [latex]{f}^{-1}\\left(x\\right)={\\frac{2 - 8x}{x}}[\/latex]<\/p>\n

25. [latex]{f}^{-1}\\left(x\\right)=\\frac{7x - 3}{1-x}[\/latex]<\/p>\n

35. [latex]{f}^{-1}\\left(x\\right)=\\sqrt{x}+4[\/latex]
\"Graph<\/p>\n

37. [latex]{f}^{-1}\\left(x\\right)=\\sqrt[3]{1-x}[\/latex]
\"Graph<\/p>\n

41. [latex]{f}^{-1}\\left(x\\right)=\\sqrt{\\frac{1}{x}}[\/latex]
\"Graph<\/p>\n

57. [latex]t\\left(h\\right)=\\sqrt{\\frac{200-h}{4.9}}[\/latex], 5.53 seconds<\/p>\n

59. [latex]r\\left(V\\right)=\\sqrt[3]{\\frac{3V}{4\\pi }}[\/latex], 3.63 feet<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<\/p><\/div>\n","protected":false},"author":13,"menu_order":5,"template":"","meta":{"_candela_citation":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":224,"module-header":"","content_attributions":null,"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/279"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/279\/revisions"}],"predecessor-version":[{"id":297,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/279\/revisions\/297"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/224"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/279\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=279"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=279"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=279"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=279"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}