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Guides d'étude > MTH 163, Precalculus

Section Exercises

1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude? 2. If a polynomial of degree n is divided by a binomial of degree 1, what is the degree of the quotient? For the following exercises, use long division to divide. Specify the quotient and the remainder. 3. (x2+5x−1)÷(x−1)\left({x}^{2}+5x - 1\right)\div \left(x - 1\right)\\ 4. (2x2−9x−5)÷(x−5)\left(2{x}^{2}-9x - 5\right)\div \left(x - 5\right)\\ 5. (3x2+23x+14)÷(x+7)\left(3{x}^{2}+23x+14\right)\div \left(x+7\right)\\ 6. (4x2−10x+6)÷(4x+2)\left(4{x}^{2}-10x+6\right)\div \left(4x+2\right)\\ 7. (6x2−25x−25)÷(6x+5)\left(6{x}^{2}-25x - 25\right)\div \left(6x+5\right)\\ 8. (−x2−1)÷(x+1)\left(-{x}^{2}-1\right)\div \left(x+1\right)\\ 9. (2x2−3x+2)÷(x+2)\left(2{x}^{2}-3x+2\right)\div \left(x+2\right)\\ 10. (x3−126)÷(x−5)\left({x}^{3}-126\right)\div \left(x - 5\right)\\ 11. (3x2−5x+4)÷(3x+1)\left(3{x}^{2}-5x+4\right)\div \left(3x+1\right)\\ 12. (x3−3x2+5x−6)÷(x−2)\left({x}^{3}-3{x}^{2}+5x - 6\right)\div \left(x - 2\right)\\ 13. (2x3+3x2−4x+15)÷(x+3)\left(2{x}^{3}+3{x}^{2}-4x+15\right)\div \left(x+3\right)\\ For the following exercises, use synthetic division to find the quotient. 14. (3x3−2x2+x−4)÷(x+3)\left(3{x}^{3}-2{x}^{2}+x - 4\right)\div \left(x+3\right)\\ 15. (2x3−6x2−7x+6)÷(x−4)\left(2{x}^{3}-6{x}^{2}-7x+6\right)\div \left(x - 4\right)\\ 16. (6x3−10x2−7x−15)÷(x+1)\left(6{x}^{3}-10{x}^{2}-7x - 15\right)\div \left(x+1\right)\\ 17. (4x3−12x2−5x−1)÷(2x+1)\left(4{x}^{3}-12{x}^{2}-5x - 1\right)\div \left(2x+1\right)\\ 18. (9x3−9x2+18x+5)÷(3x−1)\left(9{x}^{3}-9{x}^{2}+18x+5\right)\div \left(3x - 1\right)\\ 19. (3x3−2x2+x−4)÷(x+3)\left(3{x}^{3}-2{x}^{2}+x - 4\right)\div \left(x+3\right)\\ 20. (−6x3+x2−4)÷(2x−3)\left(-6{x}^{3}+{x}^{2}-4\right)\div \left(2x - 3\right)\\ 21. (2x3+7x2−13x−3)÷(2x−3)\left(2{x}^{3}+7{x}^{2}-13x - 3\right)\div \left(2x - 3\right)\\ 22. (3x3−5x2+2x+3)÷(x+2)\left(3{x}^{3}-5{x}^{2}+2x+3\right)\div \left(x+2\right)\\ 23. (4x3−5x2+13)÷(x+4)\left(4{x}^{3}-5{x}^{2}+13\right)\div \left(x+4\right)\\ 24. (x3−3x+2)÷(x+2)\left({x}^{3}-3x+2\right)\div \left(x+2\right)\\ 25. (x3−21x2+147x−343)÷(x−7)\left({x}^{3}-21{x}^{2}+147x - 343\right)\div \left(x - 7\right)\\ 26. (x3−15x2+75x−125)÷(x−5)\left({x}^{3}-15{x}^{2}+75x - 125\right)\div \left(x - 5\right)\\ 27. (9x3−x+2)÷(3x−1)\left(9{x}^{3}-x+2\right)\div \left(3x - 1\right)\\ 28. (6x3−x2+5x+2)÷(3x+1)\left(6{x}^{3}-{x}^{2}+5x+2\right)\div \left(3x+1\right)\\ 29. (x4+x3−3x2−2x+1)÷(x+1)\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\right)\div \left(x+1\right)\\ 30. (x4−3x2+1)÷(x−1)\left({x}^{4}-3{x}^{2}+1\right)\div \left(x - 1\right)\\ 31. (x4+2x3−3x2+2x+6)÷(x+3)\left({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6\right)\div \left(x+3\right)\\ 32. (x4−10x3+37x2−60x+36)÷(x−2)\left({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36\right)\div \left(x - 2\right)\\ 33. (x4−8x3+24x2−32x+16)÷(x−2)\left({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16\right)\div \left(x - 2\right)\\ 34. (x4+5x3−3x2−13x+10)÷(x+5)\left({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10\right)\div \left(x+5\right)\\ 35. (x4−12x3+54x2−108x+81)÷(x−3)\left({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81\right)\div \left(x - 3\right)\\ 36. (4x4−2x3−4x+2)÷(2x−1)\left(4{x}^{4}-2{x}^{3}-4x+2\right)\div \left(2x - 1\right)\\ 37. (4x4+2x3−4x2+2x+2)÷(2x+1)\left(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2\right)\div \left(2x+1\right)\\ For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one. 38. Factor is x2−x+3{x}^{2}-x+3\\ Graph of a polynomial that has a x-intercept at -1. 39. Factor is (x2+2x+4)\left({x}^{2}+2x+4\right)\\ Graph of a polynomial that has a x-intercept at 1. 40. Factor is x2+2x+5{x}^{2}+2x+5\\ Graph of a polynomial that has a x-intercept at 2. 41. Factor is x2+x+1{x}^{2}+x+1\\ Graph of a polynomial that has a x-intercept at 5. 42. Factor is x2+2x+2{x}^{2}+2x+2\\ Graph of a polynomial that has a x-intercept at -3. For the following exercises, use synthetic division to find the quotient and remainder. 43. 4x3−33x−2\frac{4{x}^{3}-33}{x - 2}\\ 44. 2x3+25x+3\frac{2{x}^{3}+25}{x+3}\\ 45. 3x3+2x−5x−1\frac{3{x}^{3}+2x - 5}{x - 1}\\ 46. −4x3−x2−12x+4\frac{-4{x}^{3}-{x}^{2}-12}{x+4}\\ 47. x4−22x+2\frac{{x}^{4}-22}{x+2}\\ For the following exercises, use a calculator with CAS to answer the questions. 48. Consider xk−1x−1\frac{{x}^{k}-1}{x - 1}\\ with k=1,2,3k=1, 2, 3\\. What do you expect the result to be if k = 4? 49. Consider xk+1x+1\frac{{x}^{k}+1}{x+1}\\ for k=1,3,5k=1, 3, 5\\. What do you expect the result to be if k = 7? 50. Consider x4−k4x−k\frac{{x}^{4}-{k}^{4}}{x-k}\\ for k=1,2,3k=1, 2, 3\\. What do you expect the result to be if k = 4? 51. Consider xkx+1\frac{{x}^{k}}{x+1}\\ with k=1,2,3k=1, 2, 3\\. What do you expect the result to be if k = 4? 52. Consider xkx−1\frac{{x}^{k}}{x - 1}\\ with k=1,2,3k=1, 2, 3\\. What do you expect the result to be if k = 4? For the following exercises, use synthetic division to determine the quotient involving a complex number. 53. x+1x−i\frac{x+1}{x-i}\\ 54. x2+1x−i\frac{{x}^{2}+1}{x-i}\\ 55. x+1x+i\frac{x+1}{x+i}\\ 56. x2+1x+i\frac{{x}^{2}+1}{x+i}\\ 57. x3+1x−i\frac{{x}^{3}+1}{x-i}\\ For the following exercises, use the given length and area of a rectangle to express the width algebraically. 58. Length is x+5x+5\\, area is 2x2+9x−52{x}^{2}+9x - 5\\. 59. Length is 2x + 52x\text{ }+\text{ }5\\, area is 4x3+10x2+6x+154{x}^{3}+10{x}^{2}+6x+15\\ 60. Length is 3x−43x - 4\\, area is 6x4−8x3+9x2−9x−46{x}^{4}-8{x}^{3}+9{x}^{2}-9x - 4\\ For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically. 61. Volume is 12x3+20x2−21x−3612{x}^{3}+20{x}^{2}-21x - 36\\, length is 2x+32x+3\\, width is 3x−43x - 4\\. 62. Volume is 18x3−21x2−40x+4818{x}^{3}-21{x}^{2}-40x+48\\, length is 3x−43x - 4\\, width is 3x−43x - 4\\. 63. Volume is 10x3+27x2+2x−2410{x}^{3}+27{x}^{2}+2x - 24\\, length is 5x−45x - 4\\, width is 2x+32x+3\\. 64. Volume is 10x3+30x2−8x−2410{x}^{3}+30{x}^{2}-8x - 24\\, length is 2, width is x+3x+3\\. For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically. 65. Volume is π(25x3−65x2−29x−3)\pi \left(25{x}^{3}-65{x}^{2}-29x - 3\right)\\, radius is 5x+15x+1\\. 66. Volume is π(4x3+12x2−15x−50)\pi \left(4{x}^{3}+12{x}^{2}-15x - 50\right)\\, radius is 2x+52x+5\\. 67. Volume is π(3x4+24x3+46x2−16x−32)\pi \left(3{x}^{4}+24{x}^{3}+46{x}^{2}-16x - 32\right)\\, radius is x+4x+4\\.

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