Unit 5 Test Polynomial Functions

Unit 5 Test Study Guide (Polynomial Functions) Name: ________________ Date: ________________ Per: ________________ Topic 1: Classifying Polynomials & Polynomial Operations Simplify each expression. Final answers should be written in standard form. 1. (-4m⁶n³) * (1/6)m⁻¹⁰n⁴ 256m⁸n⁴ * (1/6)m⁻¹⁰n⁴ = 128 / (3m²) 2. (8a² - 6 - 8a) + (1 - 6a - 7a²) a² - 14a - 5 3. (6x - 7x³ + 7) - (5x³ + 2x - 2x² - 1) 2x³ - 12x³ + 4x + 8 4. (y + 4)³ - 2y(y-1) (y+4)(y+4)(y+4) - 2y²+2y (y+4)(y²+8y+16) - 2y²+2y y³+8y²+16y + 4y²+32y+64 - 2y²+2y = y³+10y²+50y+64 5. (3k - 6)(k² - k + 7) = 3k³ - 3k² + 21k - 6k² + 6k - 42 = 3k³ - 9k² + 27k - 42 6. (-8c⁶d⁴ + 56c⁴d³ - 24c²d) / (8c²d) = -c⁴d³ + 7c²d² - 3 Topic 2: Factoring Polynomials Differences of Squares Sum of Cubes Differences of Cubes a² - b² = (a+b)(a-b) a³ + b³ = (a+b)(a² - ab + b²) a³ - b³ = (a-b)(a² + ab + b²) Factor each polynomial below completely. 7. 9x³ + 21x² 3x²(3x + 7) 8. 3n⁴ - 147 3(n⁴ - 49) 3(n² + 7)(n² - 7) 9. 64a³ - 343b³ (4a - 7b)(16a² + 28ab + 49b²) 10. 648w + 1029w³ 3w(216 + 343w²) 3w(6 + 7w)(36 - 42w + 49w²) 11. 32c⁶d - 162cd⁵ 2cd(16c⁵ - 81d⁴) 2cd(4c² + 9d²)(4c² - 9d) 12. 216pq - p⁷q pq(216 - p⁶) pq(6 - p²)(36 + 6p² + p⁴) 13. 2c⁵ - 2c³ - 60c 2c(c⁴ - c² - 30) 2c(c² - 6)(c² + 5) 14. 9y⁵ - 7y³ - 16 (y⁴ - 16)(y² + 9) / 9 (9y² - 16)(y² + 1) (3y + 4)(3y - 4)(y² + 1) 15. n³ + 2n² - 36n - 72 n²(n + 2) - 36(n + 2) (n² - 36)(n + 2) (n + 6)(n - 6)(n + 2) 16. 8x³ - 10x² + 28x - 35 2x²(4x - 5) + 7(4x - 5) (2x² + 7)(4x - 5) Topic 3: Graphing Polynomial Functions Graph each function and identify its key characteristics. 17. f(x) = x⁴ + 8x³ + 16x + 7 [Graph of a 'W' shaped polynomial] Domain: {R} Range: {R} Rel. Maximum(s): (-4, 7) Rel. Minimum(s): (-1.34, -2.48) End Behavior: As x -> -∞, f(x) -> -∞ As x -> ∞, f(x) -> ∞ Inc. Intervals: (-∞, -4) U (-1.34, ∞) Dec. Intervals: (-4, -1.34) 18. f(x) = -x⁴ + 3x² + 2x [Graph of a polynomial with two peaks and one valley] Domain: {R} Range: {y | y ≤ 4.85} Rel. Maximum(s): (-1, 0), (1.37, 4.85) Rel. Minimum(s): (-.36, -.35) End Behavior: As x -> -∞, f(x) -> -∞ As x -> ∞, f(x) -> -∞ Inc. Intervals: (-∞, -1), (-.36, 1.37) Dec. Intervals: (-1, -.36), (1.37, ∞) Determine the end behavior, whether the function is an even or odd degree, and the sign of the leading coefficient given each graph below. 19. [Graph of a polynomial with 3 turns, starting up and ending up] Odd degree, Negative 20. [Graph of a polynomial with 4 turns, starting down and ending down] Even degree, Positive Identify the zeros (and their multiplicity) of each function below and the effect on the graph. 21. f(x) = 3x³(x - 1)²(x - 8) | Zero | Multiplicity | Effect | | :--- | :--- | :--- | | 0 | 3 | Crosses | | 1 | 2 | Bounces | | 8 | 1 | Crosses | 22. f(x) = -x(5x + 3)³(x - 2)⁸ | Zero | Multiplicity | Effect | | :--- | :--- | :--- | | 0 | 1 | Crosses | | -3/5 | 3 | Crosses | | 2 | 8 | Bounces | 23. The graph of a polynomial function has zeros of 0 (multiplicity 2), 1 (multiplicity 2), and 5/2 (multiplicity 2). Write a function in standard form that could represent this function. f(x) = x²(x-1)²(2x-5)² f(x) = x²(x²-2x+1)(4x²-20x+25) f(x) = x²(4x⁴ - 20x³ + 25x² - 8x³ + 40x² - 50x + 4x² - 20x + 25) f(x) = x²(4x⁴ - 28x³ + 69x² - 70x + 25) f(x) = 4x⁶ - 28x⁵ + 69x⁴ - 70x³ + 25x² Topic 4: Solving Polynomial Equations Solve each equation. Simplify all irrational and complex solutions. 24. 2x³ = 48x 2x³ - 48x = 0 2x(x² - 24) = 0 2x=0 | x² = 24 x=0 | x = ±√24 | x = ± 2√6 {0, ±2√6} 25. 25x³ = 64x 25x³ - 64x = 0 x(25x² - 64) = 0 x((5x+8)(5x-8))=0 x=0 | 5x+8=0 | 5x-8=0 | x=-8/5 | x=8/5 {0, ±8/5} Topic 5: Dividing Polynomials & The Remainder Theorem Find each quotient. 32. (x³ + x² - 71x + 9) ÷ (x + 9) Long division is performed, yielding the answer: x² - 8x + 1 33. (8n³ - 36n² - 15n - 16) ÷ (n - 5) Synthetic division is performed, yielding the answer: 8n² + 4n + 5 + 9/(n-5) 34. (12a³ + 2a² - 6a - 30) ÷ (3a - 4) Long division is performed, yielding the answer: 4a² + 6a + 6 - 6/(3a-4) 35. (y⁴ + 6y³ - 4y - 31) ÷ (y + 6) Synthetic division is performed, yielding the answer: y³ - 4 - 7/(y+6) 36. Using the Remainder Theorem, find ƒ(-2) when ƒ(x) = 3x³ - 28x² + 70. Synthetic division with -2 is performed, yielding a remainder of -66. ƒ(-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) = 0.5t³ - 3t² + t + 25 t=27 Synthetic division with 27 is performed, yielding a remainder of 207425.5. $207,425,500