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Topic 8 Review TOPIC VOCABULARY Ě &RQMXJDWH5RRW7KHRUHP p. 369 Ě &RQMXJDWHV p. 369 Ě )XQGDPHQWDO7KHRUHPRI$OJHEUD p. 375 Ě UHODWLYHPLQLPXP p. 350 Ě 5HPDLQGHU7KHRUHP p. 363 Ě GHJUHHRIDPRQRPLDO p. 340 Ě PRQRPLDO p. 340 Ě URRWV p. 350 Ě GHJUHHRIDSRO\QRPLDO p. 340 Ě PXOWLSOH]HUR p. 350 Ě VWDQGDUGIRUPRIDSRO\QRPLDO Ě 'HVFDUWHVď5XOHRI6LJQV p. 370 Ě PXOWLSOLFLW\ p. 350 Ě GLIIHUHQFHRIFXEHV p. 357 Ě SRO\QRPLDO p. 340 Ě VXPRIFXEHV p. 357 Ě HQGEHKDYLRU p. 340 Ě SRO\QRPLDOIXQFWLRQ p. 340 Ě V\QWKHWLFGLYLVLRQ p. 363 Ě IDFWRUWKHRUHP p. 350 Ě 5DWLRQDO5RRW7KHRUHP p. 369 Ě WXUQLQJSRLQW p. 340 IXQFWLRQ p. 340 Ě UHODWLYHPD[LPXP p. 350 Check Your Understanding Match each vocabulary term with the description that best fits it. 1. Conjugate Root Theorem A. determines P (a) by dividing the polynomial by x - a 2. Fundamental Theorem of Algebra B. the degree equals the number of roots 3. Rational Root Theorem C. minimizes guessing fraction and integer solutions 4. Remainder Theorem D. complex numbers as roots come in pairs 8-1 Attributes of Polynomial Functions Quick Review Exercises The standard form of a polynomial function is P (x) = anx n + an-1x n-1 + c + a1x + a0, where n is a nonnegative integer and the coefficients are real numbers. A polynomial function is classified by degree. Its degree is the highest degree among its monomial term(s). The degree determines the possible number of turning points in the graph and the end behavior of the graph. Write each polynomial function in standard form, classify it by degree, and determine the end behavior of its graph. Example Classify P(x) = -4x2 + x4 by degree. How many possible turning points are there given the degree of P(x)? The degree of P(x) is 4, so its graph has either 1 or 3 turning points. 5. y = 12 - x 4 6. y = 2x 2 + 8 - 4x + x 3 7. y = x 2 + 7 - x 8. y = 10 - 3x 3 + 3x 2 + x 4 9. If the volume of a cube can be represented by a polynomial of degree 9, what is the degree of the polynomial that represents each side length? 10. A polynomial function P(x) has degree n. If n is even, is the number of turning points of the graph of P(x) even or odd? What can you say about the number of turning points if n is odd? PearsonTEXAS.com 379 8-2 Adding, Subtracting, and Multiplying Polynomials Quick Review Exercises You can add and subtract polynomials by combining like terms. Find each sum. You can multiply polynomials using the Distributive Property or the FOIL method. 11. (4x2 + 3x - 5) + (2x + 4) 12. (6x3 - x2 + 11x) + (3x3 - 9x + 6) Find each difference. Examples What is the sum of (6x2 + 3x - 8) and (5x2 - 4x)? Combine like terms. (6x2 + 3x - 8) + (5x2 - 4x) = (6x2 + 5x2) + (3x - 4x) - 8 = 11x2 - x - 8 What is the product of (4x + 5) and (2x - 1)? Use the FOIL method. Then combine like terms. (4x + 5)(2x - 1) = 8x2 - 4x + 10x - 5 13. (x2 - 8x + 7) - (2x2 - 5x - 6) 14. (12x3 - 7x2 + 4x) - (5x3 - x2 - 2) Find each product. 15. (2 - 2x2)(x + 4) 16. (2x + 1)(3x2 - x + 4) 17. (x + 5)(2x - 1)(x - 2) = 8x2 + 6x - 5 8-3 Polynomials, Linear Factors, and Zeros Quick Review Exercises For any real number a and polynomial P (x), if x - a is a factor of P (x), then a is: Write a polynomial function with the given zeros. r a zero of y = P (x) r a root (or solution) of P (x) = 0, and r an x-intercept of the graph of y = P (x). If a is a multiple zero, its multiplicity is the same as the number of times x - a appears as a factor. 18. x = -1, -1, 6 19. x = -1, 0, 2 20. x = 1, 2, 3 21. x = -2, 1, 4 Find the zeros of each function. State the multiplicity of any multiple zeros. A turning point is a relative maximum or relative minimum of a polynomial function. 22. y = 3x(x + 2) 3 23. y = x 4 - 8x 2 + 16 24. y = 4x 3 - 2x 2 - 2x 25. y = (x - 5)(x + 2) 2 Example Use a graphing calculator to find the relative maximum, relative minimum, and zeros of each function. Find the zeros for y = 3x 3 - 6x 2 + 3x and state the multiplicity of any multiple zeros. 26. f (x) = x 4 - 5x 3 + 5x 2 - 3 y = 3x(x2 - 2x + 1) Factor out the GCF, 3x. 27. f (x) = 5x 3 + x 2 - 9x + 4 y = 3x(x - 1)(x - 1) Factor the quadratic. 28. f (x) = x 4 - 4x - 1 The zeros are 0, and 1 with multiplicity 2. 380 Topic 8 Review 29. f (x) = x 3 - 3x 2 - 3x - 4 8-4 Solving Polynomial Equations Quick Review Exercises One way to solve a polynomial equation is by factoring. First write the equation in the form P (x) = 0, where P (x) is the polynomial. Then factor the polynomial. Last, use the Zero-Product Property to find the solutions, or roots. The solutions may be real or imaginary. Real solutions and approximations of irrational solutions can also be found by using a graphing calculator. Find the real or imaginary solutions of each equation by factoring. 30. x 2 - 11x = -24 31. 4x 2 = -4x - 1 32. 3x 3 + 3x 2 = 27x 33. 2x 2 + 3 = 4x Find the real roots of each equation by graphing. 34. x 4 + 3x 2 - 2x + 5 = 0 Example Solve x3 x3 + + 35. x 2 + 3 = x 3 - 5 4x 2 4x 2 = 12x by factoring. - 12x = 0 Subtract 12x from each side. x(x - 2)(x + 6) = 0 Factor the left side. x = 0, x - 2 = 0, x + 6 = 0 Zero-Product Property x = 0, x = 2, x = - 6 Solve each equation. 36. The height and width of a rectangular prism are each 2 inches shorter than the length of the prism. The volume of the prism is 40 cubic inches. Approximate the dimensions of the prism to the nearest hundredth. The solutions are 0, 2, and -6. 8-5 Dividing Polynomials Quick Review Exercises You can divide a polynomial by one of its factors to find another factor. When you divide by a linear factor, you can simplify this division by writing only the coefficients of each term. This is called synthetic division. The Remainder Theorem says that P (a) is the remainder when you divide P (x) by x - a. Divide using long division. Check your answers. 37. (x 3 + 7x 2 + 15x + 9) , (x + 1) 38. (2x 3 - 7x 2 - 7x + 13) , (x - 4) Determine whether each binomial is a factor of x 3 + x 2 − 10x + 8. Example 39. x - 2 Let P (x) = 3x 2 - 13x + 15. What is P (3)? Divide using synthetic division. According to the Remainder Theorem, P (3) is the remainder when you divide P (x) by x - 3. 41. (x 3 + 5x 2 - x - 5) , (x + 5) 3 3 ⫺13 15 9 ⫺12 3 ⫺4 Put the opposite of the constant in the divisor at the top left. 3 The quotient is 3x - 4 with remainder 3, so P (3) = 3. 40. x - 4 42. (2x 3 + 14x 2 - 58x) , (x + 10) 43. (5x 3 + 8x 2 - 60) , (x - 2) Use the Remainder Theorem to determine the value of P (a). 44. P (x) = 2x 3 + 5x 2 + 7x - 4, a = -2 45. P (x) = x 3 - 4x 2 + 2x + 3, a = 1 PearsonTEXAS.com 381 8-6 Theorems About Roots of Polynomial Equations Quick Review Exercises The Rational Root Theorem gives a way to determine the possible roots of a polynomial equation P (x) = 0. If the coefficients of P(x) are all integers, then every root of the p equation can be written in the form q , where p is a factor of the constant term and q is a factor of the leading coefficient. List the possible rational roots of P (x) given by the Rational Root Theorem. The Conjugate Root Theorem states that if P(x) is a polynomial with rational coefficients, then irrational roots that have the form a + 1b and imaginary roots of P (x) = 0 come in conjugate pairs. Therefore, if a + 1b is an irrational root, where a and b are rational, then a - 1b is also a root. Likewise, if a + bi is a root, where a and b are real and i is the imaginary unit, then a - bi is also a root. 48. P (x) = 4x 4 - 2x 3 + x 2 - 12 Descartes’ Rule of Signs gives a way to determine the possible number of positive and negative real roots by analyzing the signs of the coefficients. The number of positive real roots is equal to the number of sign changes in consecutive coefficients of P(x), or is less than that by an even number. The number of negative real roots is equal to the number of sign changes in consecutive coefficients of P( -x), or is less than that by an even number. 46. P(x) = x 3 + 4x 2 - 10x + 6 47. P (x) = 3x 3 - x 2 - 7x + 2 49. P (x) = 3x 4 - 4x 3 - x 2 - 7 Find any rational roots of P (x). 50. P(x) = x 3 + 2x 2 + 4x + 21 51. P(x) = x 3 + 5x 2 + x + 5 52. P(x) = 2x 3 + 7x 2 - 5x - 4 53. P(x) = 3x 4 + 2x 3 - 9x 2 + 4 A polynomial P (x) has rational coefficients. Name additional roots of P (x) given the following roots. 54. 1 - i and 5 55. 5 + 13 and - 12 Example 56. -3i and 7i Find the rational roots of P (x) = 0 if P (x) = 2x 3 - 4x 2 - 10x + 12. 57. -2 + 211 and -4 - 6i List the possible roots: { 12, {1, { 32, {2, {3, {4, {6, {12. Use synthetic division to test roots. Write a polynomial function with the given roots. 3 2 2 ⫺4 ⫺10 12 6 6 ⫺12 2 ⫺4 0 So x - 3 and (2x 2 + 2x - 4) are factors of P(x). P (x) = (x - 3)(2x 2 + 2x - 4) Factor the quadratic. P (x) = 2(x - 3)(x + 2)(x - 1) Solve 2(x - 3)(x + 2)(x - 1) = 0. x = 3, x = - 2, or x = 1 The rational roots are 3, -2, and 1. 382 Topic 8 5HYLHZ 58. 7 and 10 59. -3 and 5i 60. 6 - i 61. 3 + i, 2, and -4 Determine the possible number of positive real zeros and negative real zeros for each polynomial function given by Descartes’ Rule of Signs. 62. P (x) = 5x 3 + 7x 2 - 2x - 1 63. P (x) = -3x 3 + 11x 2 + 12x - 8 64. P (x) = 6x 4 - x 3 + 5x 2 - x + 9 65. P (x) = -x 4 - 3x 3 + 8x 2 + 2x - 14 8-7 The Fundamental Theorem of Algebra Quick Review Exercises The Fundamental Theorem of Algebra states that if P (x) is a polynomial of degree n, where n Ú 1, then P (x) = 0 has exactly n roots. This includes multiple and complex roots. Find the number of roots for each equation. Example 68. -x 5 - 6 = 0 Use the Fundamental Theorem of Algebra to determine the number of roots for x 4 + 2x 2 - 3 = 0. 69. 5x 4 - 7x 6 + 2x 3 + 8x 2 + 4x - 11 = 0 Because the polynomial is of degree 4, it has 4 roots. 66. x 3 - 2x + 5 = 0 67. 2 - x 4 + x 2 = 0 Find all the zeros for each function. 70. P (x) = x 3 + 5x 2 - 4x - 2 71. P (x) = x 4 - 4x 3 - x 2 + 20x - 20 72. P (x) = 2x 3 - 3x 2 + 3x - 2 73. P (x) = x 4 - 4x 3 - 16x 2 + 21x + 18 PearsonTEXAS.com 383