{"id":2115,"date":"2025-08-04T18:11:58","date_gmt":"2025-08-04T18:11:58","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2115"},"modified":"2026-08-05T18:07:25","modified_gmt":"2026-08-05T18:07:25","slug":"polynomial-equations-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polynomial-equations-get-stronger\/","title":{"raw":"Polynomial Equations: Get Stronger","rendered":"Polynomial Equations: Get Stronger"},"content":{"raw":"

Dividing Polynomials<\/h2>\r\n1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?\r\n\r\nFor the following exercises, use long division to divide. Specify the quotient and the remainder.\r\n\r\n2. [latex]\\left({x}^{2}+5x - 1\\right)\\div \\left(x - 1\\right)[\/latex]\r\n\r\n3. [latex]\\left(6{x}^{2}-25x - 25\\right)\\div \\left(6x+5\\right)[\/latex]\r\n\r\n4. [latex]\\left(2{x}^{3}+3{x}^{2}-4x+15\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\nFor the following exercises, use synthetic division to find the quotient.\r\n\r\n5. [latex]\\left(2{x}^{3}-6{x}^{2}-7x+6\\right)\\div \\left(x - 4\\right)[\/latex]\r\n\r\n6. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n7. [latex]\\left(4{x}^{3}-5{x}^{2}+13\\right)\\div \\left(x+4\\right)[\/latex]\r\n\r\n8. [latex]\\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\\right)\\div \\left(x+1\\right)[\/latex]\r\n\r\nFor the following exercises, use synthetic division to find the quotient and remainder.\r\n\r\n9. [latex]\\frac{4{x}^{3}-33}{x - 2}[\/latex]\r\n\r\n10. [latex]\\frac{3{x}^{3}+2x - 5}{x - 1}[\/latex]\r\n\r\nFor the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.\r\n\r\n11. Volume is [latex]12{x}^{3}+20{x}^{2}-21x - 36[\/latex], length is [latex]2x+3[\/latex], width is [latex]3x - 4[\/latex].\r\n\r\n12. Volume is [latex]10{x}^{3}+27{x}^{2}+2x - 24[\/latex], length is [latex]5x - 4[\/latex], width is [latex]2x+3[\/latex].\r\n

Complex Numbers<\/h2>\r\n1. Explain how to add complex numbers.\r\n\r\n2. Give an example to show the product of two imaginary numbers is not always imaginary.\r\n\r\nFor the following exercises, evaluate the algebraic expressions.\r\n\r\n3. [latex]\\text{If }f\\left(x\\right)={x}^{2}+3x+5[\/latex], evaluate [latex]f\\left(2+i\\right)[\/latex].\r\n\r\n4. [latex]\\text{If }f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex], evaluate [latex]f\\left(5i\\right)[\/latex].\r\n\r\nFor the following exercises, plot the complex numbers on the complex plane.\r\n\r\n5. [latex]1 - 2i[\/latex]\r\n\r\n6. i<\/em>\r\n\r\nFor the following exercises, perform the indicated operation and express the result as a simplified complex number.\r\n\r\n7. [latex]\\left(3+2i\\right)+\\left(5 - 3i\\right)[\/latex]\r\n\r\n8. [latex]\\left(-5+3i\\right)-\\left(6-i\\right)[\/latex]\r\n\r\n9. [latex]\\left(5 - 2i\\right)\\left(3i\\right)[\/latex]\r\n\r\n10. [latex]\\left(-2+4i\\right)\\left(8\\right)[\/latex]\r\n\r\n11. [latex]\\left(-1+2i\\right)\\left(-2+3i\\right)[\/latex]\r\n\r\n12. [latex]\\left(3+4i\\right)\\left(3 - 4i\\right)[\/latex]\r\n\r\n13. [latex]\\frac{6+4i}{i}[\/latex]\r\n\r\n14. [latex]\\frac{3+4i}{2-i}[\/latex]\r\n\r\n15. [latex]\\sqrt{-9}+3\\sqrt{-16}[\/latex]\r\n\r\n16. [latex]\\frac{2+\\sqrt{-12}}{2}[\/latex]\r\n\r\n17. [latex]{i}^{8}[\/latex]\r\n\r\n18. [latex]{i}^{22}[\/latex]\r\n

Zeros of Polynomial Functions<\/h2>\r\n1. What is the difference between rational and real zeros?\r\n\r\n2. If synthetic division reveals a zero, why should we try that value again as a possible solution?\r\n\r\nFor the following exercises, use the Remainder Theorem to find the remainder.\r\n\r\n3. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]\r\n\r\n4. [latex]\\left({x}^{4}-1\\right)\\div \\left(x - 4\\right)[\/latex]\r\n\r\n5. [latex]\\left(4{x}^{3}+5{x}^{2}-2x+7\\right)\\div \\left(x+2\\right)[\/latex]\r\n\r\nFor the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.\r\n\r\n6. [latex]f\\left(x\\right)=2{x}^{3}+{x}^{2}-5x+2;\\text{ }x+2[\/latex]\r\n\r\n7. [latex]2{x}^{3}+5{x}^{2}-12x - 30,\\text{ }2x+5[\/latex]\r\n\r\nFor the following exercises, use the Rational Zero Theorem to find all real zeros.\r\n\r\n8. [latex]2{x}^{3}+7{x}^{2}-10x - 24=0[\/latex]\r\n\r\n9. [latex]{x}^{3}+5{x}^{2}-16x - 80=0[\/latex]\r\n\r\n10. [latex]2{x}^{3}-3{x}^{2}-x+1=0[\/latex]\r\n\r\n11. [latex]2{x}^{3}-5{x}^{2}+9x - 9=0[\/latex]\r\n\r\n12. [latex]{x}^{4}-2{x}^{3}-7{x}^{2}+8x+12=0[\/latex]\r\n\r\n13. [latex]{x}^{4}+2{x}^{3}-4{x}^{2}-10x - 5=0[\/latex]\r\n\r\nFor the following exercises, find all complex solutions (real and non-real).\r\n\r\n14. [latex]{x}^{3}-8{x}^{2}+25x - 26=0[\/latex]\r\n\r\n15. [latex]2{x}^{3}-3{x}^{2}+32x+17=0[\/latex]\r\n\r\nFor the following exercises, list all possible rational zeros for the functions.\r\n\r\n16. [latex]f\\left(x\\right)=2{x}^{{}^{3}}+3{x}^{2}-8x+5[\/latex]\r\n\r\n17. [latex]f\\left(x\\right)=6{x}^{4}-10{x}^{2}+13x+1[\/latex]\r\n\r\nFor the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.\r\n\r\n18. [latex]f\\left(x\\right)=6{x}^{3}-7{x}^{2}+1[\/latex]\r\n\r\n19. [latex]f\\left(x\\right)=8{x}^{3}-6{x}^{2}-23x+6[\/latex]\r\n\r\n20. [latex]f\\left(x\\right)=16{x}^{4}-24{x}^{3}+{x}^{2}-15x+25[\/latex]\r\n\r\nFor the following exercises, find the dimensions of the box described.\r\n\r\n21. The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.\r\n\r\n22. The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.\r\n

Inverse and Radical Functions<\/h2>\r\n1. Explain why we cannot find inverse functions for all polynomial functions.\r\n\r\n2. When finding the inverse of a radical function, what restriction will we need to make?\r\n\r\nFor the following exercises, find the inverse of the function on the given domain.\r\n\r\n3. [latex]f\\left(x\\right)={\\left(x - 4\\right)}^{2}, \\left[4,\\infty \\right)[\/latex]\r\n\r\n4. [latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-3, \\left[-1,\\infty \\right)[\/latex]\r\n\r\n5. [latex]f\\left(x\\right)=3{x}^{2}+5,\\left(-\\infty ,0\\right],\\left[0,\\infty \\right)[\/latex]\r\n\r\n6. [latex]f\\left(x\\right)=9-{x}^{2}, \\left[0,\\infty \\right)[\/latex]\r\n\r\n7. [latex]f\\left(x\\right)={x}^{2}-6x+3, \\left[3,\\infty \\right)[\/latex]\r\n\r\nFor the following exercises, find the inverse of the functions.\r\n\r\n8. [latex]f\\left(x\\right)={x}^{3}+5[\/latex]\r\n\r\n9. [latex]f\\left(x\\right)=\\sqrt{2x+1}[\/latex]\r\n\r\n10. [latex]f\\left(x\\right)=9+2\\sqrt[3]{x}[\/latex]\r\n\r\n11. [latex]f\\left(x\\right)=\\frac{2}{x+8}[\/latex]\r\n\r\n12. [latex]f\\left(x\\right)=\\frac{x+3}{x+7}[\/latex]\r\n\r\nFor the following exercises, find the inverse of the function and graph both the function and its inverse.\r\n\r\n13. [latex]f\\left(x\\right)={\\left(x - 4\\right)}^{2},x\\ge 4[\/latex]\r\n\r\n14. [latex]f\\left(x\\right)=1-{x}^{3}[\/latex]\r\n\r\n15. [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}},x\\ge 0[\/latex]\r\n\r\nFor the following exercises, determine the function described and then use it to answer the question.\r\n\r\n16. An object dropped from a height of 200 meters has a height, [latex]h\\left(t\\right)[\/latex], in meters after t<\/em> seconds have lapsed, such that [latex]h\\left(t\\right)=200 - 4.9{t}^{2}[\/latex]. Express t<\/em> as a function of height, h<\/em>, and find the time to reach a height of 50 meters.\r\n\r\n17. The volume, V<\/em>, of a sphere in terms of its radius, r<\/em>, is given by [latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]. Express r<\/em> as a function of V<\/em>, and find the radius of a sphere with volume of 200 cubic feet.","rendered":"

Dividing Polynomials<\/h2>\n

1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?<\/p>\n

For the following exercises, use long division to divide. Specify the quotient and the remainder.<\/p>\n

2. [latex]\\left({x}^{2}+5x - 1\\right)\\div \\left(x - 1\\right)[\/latex]<\/p>\n

3. [latex]\\left(6{x}^{2}-25x - 25\\right)\\div \\left(6x+5\\right)[\/latex]<\/p>\n

4. [latex]\\left(2{x}^{3}+3{x}^{2}-4x+15\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n

For the following exercises, use synthetic division to find the quotient.<\/p>\n

5. [latex]\\left(2{x}^{3}-6{x}^{2}-7x+6\\right)\\div \\left(x - 4\\right)[\/latex]<\/p>\n

6. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n

7. [latex]\\left(4{x}^{3}-5{x}^{2}+13\\right)\\div \\left(x+4\\right)[\/latex]<\/p>\n

8. [latex]\\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\\right)\\div \\left(x+1\\right)[\/latex]<\/p>\n

For the following exercises, use synthetic division to find the quotient and remainder.<\/p>\n

9. [latex]\\frac{4{x}^{3}-33}{x - 2}[\/latex]<\/p>\n

10. [latex]\\frac{3{x}^{3}+2x - 5}{x - 1}[\/latex]<\/p>\n

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.<\/p>\n

11. Volume is [latex]12{x}^{3}+20{x}^{2}-21x - 36[\/latex], length is [latex]2x+3[\/latex], width is [latex]3x - 4[\/latex].<\/p>\n

12. Volume is [latex]10{x}^{3}+27{x}^{2}+2x - 24[\/latex], length is [latex]5x - 4[\/latex], width is [latex]2x+3[\/latex].<\/p>\n

Complex Numbers<\/h2>\n

1. Explain how to add complex numbers.<\/p>\n

2. Give an example to show the product of two imaginary numbers is not always imaginary.<\/p>\n

For the following exercises, evaluate the algebraic expressions.<\/p>\n

3. [latex]\\text{If }f\\left(x\\right)={x}^{2}+3x+5[\/latex], evaluate [latex]f\\left(2+i\\right)[\/latex].<\/p>\n

4. [latex]\\text{If }f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex], evaluate [latex]f\\left(5i\\right)[\/latex].<\/p>\n

For the following exercises, plot the complex numbers on the complex plane.<\/p>\n

5. [latex]1 - 2i[\/latex]<\/p>\n

6. i<\/em><\/p>\n

For the following exercises, perform the indicated operation and express the result as a simplified complex number.<\/p>\n

7. [latex]\\left(3+2i\\right)+\\left(5 - 3i\\right)[\/latex]<\/p>\n

8. [latex]\\left(-5+3i\\right)-\\left(6-i\\right)[\/latex]<\/p>\n

9. [latex]\\left(5 - 2i\\right)\\left(3i\\right)[\/latex]<\/p>\n

10. [latex]\\left(-2+4i\\right)\\left(8\\right)[\/latex]<\/p>\n

11. [latex]\\left(-1+2i\\right)\\left(-2+3i\\right)[\/latex]<\/p>\n

12. [latex]\\left(3+4i\\right)\\left(3 - 4i\\right)[\/latex]<\/p>\n

13. [latex]\\frac{6+4i}{i}[\/latex]<\/p>\n

14. [latex]\\frac{3+4i}{2-i}[\/latex]<\/p>\n

15. [latex]\\sqrt{-9}+3\\sqrt{-16}[\/latex]<\/p>\n

16. [latex]\\frac{2+\\sqrt{-12}}{2}[\/latex]<\/p>\n

17. [latex]{i}^{8}[\/latex]<\/p>\n

18. [latex]{i}^{22}[\/latex]<\/p>\n

Zeros of Polynomial Functions<\/h2>\n

1. What is the difference between rational and real zeros?<\/p>\n

2. If synthetic division reveals a zero, why should we try that value again as a possible solution?<\/p>\n

For the following exercises, use the Remainder Theorem to find the remainder.<\/p>\n

3. [latex]\\left(3{x}^{3}-2{x}^{2}+x - 4\\right)\\div \\left(x+3\\right)[\/latex]<\/p>\n

4. [latex]\\left({x}^{4}-1\\right)\\div \\left(x - 4\\right)[\/latex]<\/p>\n

5. [latex]\\left(4{x}^{3}+5{x}^{2}-2x+7\\right)\\div \\left(x+2\\right)[\/latex]<\/p>\n

For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.<\/p>\n

6. [latex]f\\left(x\\right)=2{x}^{3}+{x}^{2}-5x+2;\\text{ }x+2[\/latex]<\/p>\n

7. [latex]2{x}^{3}+5{x}^{2}-12x - 30,\\text{ }2x+5[\/latex]<\/p>\n

For the following exercises, use the Rational Zero Theorem to find all real zeros.<\/p>\n

8. [latex]2{x}^{3}+7{x}^{2}-10x - 24=0[\/latex]<\/p>\n

9. [latex]{x}^{3}+5{x}^{2}-16x - 80=0[\/latex]<\/p>\n

10. [latex]2{x}^{3}-3{x}^{2}-x+1=0[\/latex]<\/p>\n

11. [latex]2{x}^{3}-5{x}^{2}+9x - 9=0[\/latex]<\/p>\n

12. [latex]{x}^{4}-2{x}^{3}-7{x}^{2}+8x+12=0[\/latex]<\/p>\n

13. [latex]{x}^{4}+2{x}^{3}-4{x}^{2}-10x - 5=0[\/latex]<\/p>\n

For the following exercises, find all complex solutions (real and non-real).<\/p>\n

14. [latex]{x}^{3}-8{x}^{2}+25x - 26=0[\/latex]<\/p>\n

15. [latex]2{x}^{3}-3{x}^{2}+32x+17=0[\/latex]<\/p>\n

For the following exercises, list all possible rational zeros for the functions.<\/p>\n

16. [latex]f\\left(x\\right)=2{x}^{{}^{3}}+3{x}^{2}-8x+5[\/latex]<\/p>\n

17. [latex]f\\left(x\\right)=6{x}^{4}-10{x}^{2}+13x+1[\/latex]<\/p>\n

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.<\/p>\n

18. [latex]f\\left(x\\right)=6{x}^{3}-7{x}^{2}+1[\/latex]<\/p>\n

19. [latex]f\\left(x\\right)=8{x}^{3}-6{x}^{2}-23x+6[\/latex]<\/p>\n

20. [latex]f\\left(x\\right)=16{x}^{4}-24{x}^{3}+{x}^{2}-15x+25[\/latex]<\/p>\n

For the following exercises, find the dimensions of the box described.<\/p>\n

21. The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.<\/p>\n

22. The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.<\/p>\n

Inverse and Radical Functions<\/h2>\n

1. Explain why we cannot find inverse functions for all polynomial functions.<\/p>\n

2. When finding the inverse of a radical function, what restriction will we need to make?<\/p>\n

For the following exercises, find the inverse of the function on the given domain.<\/p>\n

3. [latex]f\\left(x\\right)={\\left(x - 4\\right)}^{2}, \\left[4,\\infty \\right)[\/latex]<\/p>\n

4. [latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-3, \\left[-1,\\infty \\right)[\/latex]<\/p>\n

5. [latex]f\\left(x\\right)=3{x}^{2}+5,\\left(-\\infty ,0\\right],\\left[0,\\infty \\right)[\/latex]<\/p>\n

6. [latex]f\\left(x\\right)=9-{x}^{2}, \\left[0,\\infty \\right)[\/latex]<\/p>\n

7. [latex]f\\left(x\\right)={x}^{2}-6x+3, \\left[3,\\infty \\right)[\/latex]<\/p>\n

For the following exercises, find the inverse of the functions.<\/p>\n

8. [latex]f\\left(x\\right)={x}^{3}+5[\/latex]<\/p>\n

9. [latex]f\\left(x\\right)=\\sqrt{2x+1}[\/latex]<\/p>\n

10. [latex]f\\left(x\\right)=9+2\\sqrt[3]{x}[\/latex]<\/p>\n

11. [latex]f\\left(x\\right)=\\frac{2}{x+8}[\/latex]<\/p>\n

12. [latex]f\\left(x\\right)=\\frac{x+3}{x+7}[\/latex]<\/p>\n

For the following exercises, find the inverse of the function and graph both the function and its inverse.<\/p>\n

13. [latex]f\\left(x\\right)={\\left(x - 4\\right)}^{2},x\\ge 4[\/latex]<\/p>\n

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

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

For the following exercises, determine the function described and then use it to answer the question.<\/p>\n

16. An object dropped from a height of 200 meters has a height, [latex]h\\left(t\\right)[\/latex], in meters after t<\/em> seconds have lapsed, such that [latex]h\\left(t\\right)=200 - 4.9{t}^{2}[\/latex]. Express t<\/em> as a function of height, h<\/em>, and find the time to reach a height of 50 meters.<\/p>\n

17. The volume, V<\/em>, of a sphere in terms of its radius, r<\/em>, is given by [latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]. Express r<\/em> as a function of V<\/em>, and find the radius of a sphere with volume of 200 cubic feet.<\/p>\n","protected":false},"author":67,"menu_order":28,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":506,"module-header":"- Select Header -","content_attributions":[],"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2115"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2115\/revisions"}],"predecessor-version":[{"id":6342,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2115\/revisions\/6342"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/506"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2115\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2115"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2115"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2115"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}