Unit 5 Test Study Guide (Polynomial Functions) PDF

## Unit 5 Test Study Guide (Polynomial Functions) ### Topic 1: Classifying Polynomials & Polynomial Operations **Simplify each expression. Final answers should be written in standard form.** 1. (-4m²n)^1/6 * (1/m^10 * n^4)/ (256m^8n^4 * m^2 * n^4)^1/3 = -4m^2n * n^4/m^10 * (256m^10n^8)^1/3...

## Unit 5 Test Study Guide (Polynomial Functions) ### Topic 1: Classifying Polynomials & Polynomial Operations **Simplify each expression. Final answers should be written in standard form.** 1. (-4m²n)^1/6 * (1/m^10 * n^4)/ (256m^8n^4 * m^2 * n^4)^1/3 = -4m^2n * n^4/m^10 * (256m^10n^8)^1/3 = -4m^2n^5/m^10 * 4m^10n^8/3 = -16m^10n^13/3m^10 = -16n^13/3 2. (8a^2 - 6 - 8a) + (1 - 6a - 7a^2) = 8a^2 - 6 - 8a + 1 - 6a - 7a^2 = a^2 - 14a - 5 3. (6x - 7x^2 + 7) - (5x^2 + 2x - 2x^3 - 1) = 6x - 7x^2 + 7 - 5x^2 - 2x + 2x^3 + 1 = 2x^3 - 12x^2 + 4x + 8 4. (y + 4)^3 - 2y(y - 1) = (y + 4)(y + 4)(y + 4) - 2y^2 + 2y = (y^2 + 8y + 16)(y + 4) - 2y^2 + 2y = y^3 + 8y^2 + 16y + 4y^2 + 32y + 64 - 2y^2 + 2y = y^3 + 10y^2 + 50y + 64 5. (3k - 6)(k^2 - k + 7) = 3k^3 - 3k^2 + 21k - 6k^2 + 6k - 42 = 3k^3 - 9k^2 + 27k - 42 6. -8c^6d + 56c^4d^2 - 24c^2d^3 8c^2d(-c^4d^2 + 7c^2d - 3) ### Topic 2: Factoring Polynomials **Differences of Squares** a^2 - b^2 = (a + b)(a - b) **Sum of Cubes** a^3 + b^3 = (a + b)(a^2 -ab + b^2) **Differences of Cubes** a^3 - b^3 = (a - b)(a^2 + ab + b^2) **Factor each polynomial below completely.** 7. 9x^3 + 21x^2 = 3x^2(3x + 7) 8. 3n^4 - 147 = 3(n^4 - 49) = 3(n^2 + 7)(n^2 - 7) 9. 64a^3 - 343b^3 = (4a - 7b)(16a^2 + 28ab + 49b^2) 10. 648w^4 + 1029w = 3w(216 + 343w^3) = 3w(6 + 7w)(36 - 42w + 49w^2) 11. 32c^5d - 162cd^3 = 2cd(16c^4 - 81d^2) = 2cd(4c^2 + 9d)(4c^2 - 9d) 12. 216pq^7 - pq = pq(216 - p^6) = pq(6 - p^2)(36 + 6p^2 + p^4) 13. 2c^5 - 2c^3 - 60c = 2c(c^4 - c^2 - 30) = 2c(c^2 - 6)(c^2 + 5) 14. 9y^4 - 7y^2 - 16 = (y^2 + 1)(9y^2 - 16) = (y^2 + 1)(3y + 4)(3y - 4) 15. n^3 + 2n^2 - 36n - 72 = n^2(n + 2) - 36(n + 2) = (n^2 - 36)(n + 2) = (n + 6)(n - 6)(n + 2) 16. 8x^3 - 10x^2 + 28x - 35 = 2x^2(4x - 5) + 7(4x - 5) = (2x^2 + 7)(4x - 5) ### Topic 3: Graphing Polynomial Functions **Graph each function and identify its key characteristics.** 17. f(x) = x^4 + 8x^2 + 16x + 7 **Domain:** (-∞, ∞) **Range:** (-∞, ∞) **Rel. Maximum(s):** (-4, 7) **Rel. Minimum(s):** (-1.34, -2.48) **End Behavior:** As x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞ **Inc. Intervals:** (-∞, -4) U (-1.34, ∞) **Dec. Intervals:** (-4, -1.34) 18. f(x) = x^4 + 3x^3 + 2x **Domain:** (-∞, ∞) **Range:** (-∞, 4.85) **Rel. Maximum(s):** (-1, 0) U (1.37, 4.85) **Rel. Minimum(s):** (-.36, -.35) **End Behavior:** As x → ∞, f(x) → ∞ and as x → - ∞, f(x) → -∞ **Inc. Intervals:** (-∞, -1) U (-.36, 1.37) **Dec. Intervals:** (-1, -.36) U (1.37, ∞) ### Topic 4: Solving Polynomial Equations **Solve each equation. Simplify all irrational and complex solutions.** 24. 2x^4 - 48x^2 = 0 = 2x^2(x^2 - 24) = 2x^2 = 0, x^2 - 24 = 0 = x = 0, x^2 = 24 = x = 0, x = ± √24 = x = 0, x = ± 2√6 25. 25x^3 = 64x = 25x^3 - 64x = 0 = x(25x^2 - 64) = 0 = x = 0, 25x^2 - 64 = 0 = x = 0, x^2 = 64/25 = x = 0, x = ± √(64/25) = x = 0, x = ± 8/5 ### Topic 5: Dividing Polynomials & The Remainder Theorem **Find each quotient.** 32. (x^3 + x^2 - 71x + 9) ÷ ( x + 9) ``` x^2 - 8x + 1 x + 9 | x^3 + x^2 - 71x + 9 -(x^3 + 9x^2) -8x^2 - 71x -(-8x^2 - 72x) x + 9 -( x + 9) 0 x^2 - 8x + 1 ``` 33. (8n^3 - 36n^2 - 15n - 16) ÷ (n - 5) ``` 8n^2 + 4n + 5 + 9/(n - 5) n - 5 | 8n^3 - 36n^2 - 15n - 16 -(8n^3 - 40n^2) 4n^2 - 15n -(4n^2 - 20n) 5n - 16 -(5n - 25) 9 8n^2 + 4n + 5 + 9/(n - 5) ``` 34. (12a^2 + 2a^2 - 6a - 30) ÷ (3a - 4) ``` 4a^2 + 6a + 6 3a -4 | 12a^3 + 2a^2 - 6a - 30 -(12a^3 - 16a^2 ) 18a^2 - 6a -(18a^2 - 24a) 18a - 30 -( 18a - 24) -6 4a^2 + 6a + 6 ``` 35. (y^4 + 6y^3 - 4y - 31) ÷ (y + 6) ``` y^3 - 4 y+6 | y^4 + 6y^3 - 4y - 31 -(y^4 + 6y^3) 0 - 4y -(-2y^3) -4y - 31 -(-4y - 24) -7 y^3 - 4 ``` 36. Using the Remainder Theorem, find f(-2) when f(x) = 3x^4 - 28x^2 + 70 ``` 3 -28 0 70 -2 | 3 -28 0 70 -6 68 -136 3 -34 68 -66 f(-2) = -66 ``` 37. The profit P of a small business (in thousands of dollars) since it was founded can be modeled by the function below, where t is the years since 1990. Use the Remainder Theorem to find the company's profit in 2017. P(t) = 0.5t^4 - 3t^3 + t^2 + 25 ``` 0.5 -3 1 25 27 | 0.5 -3 1 25 13.5 283.5 7681.5 0.5 10.5 284.5 7681.5 207400.5 0.5 10.5 284.5 7681.5 207425.5 $207,425,500 ``` ### Topic 6: Operations & Compositions of Functions Given f(x) = x^2 + 4x - 12, g(x) = 5x^2 - 2, and h(x) = x + 7, find each function. Indicate any restrictions in the domain. 38. (f - g)(x) = (x^2 + 4x - 12) - (5x^2 - 2) = -4x^2 + 4x - 10 39. (h * g)(x) = (x + 7)(5x^2 - 2) = 5x^3 + 35x^2 - 2x - 14 40. (f / h)(x) = (x^2 + 4x - 12) / (x + 7) ; {x|x ≠ -7, 2, 6} 41. (g o h)(x) = 5(x + 7)^2 - 2 = 5(x^2 + 14x + 49) - 2 = 5x^2 + 70x + 243 **Use the same functions above, evaluate each function.** 42. (g + h)(-4) = g(-4) + h(-4) = 5(-4)^ 2 - 2 + (-4) + 7 = 81 43. (h o f)(2) = h(f(2)) = h(2^2 + 4(2) - 12) = h(0) = 0 + 7 = 7 ### Topic 7: Regression 44. The population present in a bacteria culture over 5 days is given in the table below. Write a cubic function to represent the data. Time (days) | Population :---|:--- 0 | 28 1 | 135 2 | 219 3 | 332 4 | 520 5 | 834 f(x) = 8.18x^3 - 35.41x^2 + 133.83x + 28.08 45. Use a cubic function to estimate the value of y when x is -8. How does the estimate change when a quartic function is used instead? x | y :---|:--- -4 | 975 0 | 128 4 | -9 8 | -160 12 | -893 **Cubic:** f(x) = -1.68x^3 + 21.14x^2 - 97.48x + 136.91 f(-8) = 3129.87 **Quartic:** f(x) = 0.03x^4 - 2.09x^3 + 21.78x^2 - 89.58x + 128 f(-8) = 3431.52 The estimate differs by 301.65.