Math Polynomials Review Solutions

Algebra 2 Polynomials Test Review Name: Key Date: Simplify the expression. 1. (-b³c⁴)⁵ (-b)¹⁵c²⁰ 2. d²(3de⁵d²) 3d³e⁵d⁴ 3. 3x(-2xyz)² 3x(4x²y²z²) = 12x³y²z² 4. (4m²-3m-3)-(3m²-3m-3) m² - 2e - 3 + 3m + 3 m² + 2e 5. (x+b)(x-b)-(x+b)³ x² - b² - (x³ + 3x²b + 3xb² + b³) x² - b² - x³ - 3x²b - 3xb² - b³ -x³ - 2x²b - 4xb² - b³ 6. (2x²-5x+2)/(2x²+x-6) (2x-1)(x-2)/(2x+3)(x-2) (2x-1)/(2x+3) 7. (m⁷w⁻¹)/(w⁵) * (w³)/(m²) w²/m⁵ * w³ / m² = w⁵ / m⁷ 8. (2⁻¹tu⁻¹)²(6tu⁻¹)² 2⁻³t²u⁻² * 6⁻²t⁻²u² t⁴ / 2³6²u⁴ t⁴ / 288u⁴ 9. (2x⁻⁵(3xᵃ⁺²)³) / (a-3) 54xᵃ⁺¹ / (a-3) Solve the following equations. 10. (3⁻¹)²ˣ = (1/27)ˣ⁻⁵ 3⁻²ˣ = 3⁻³ˣ⁺¹⁵ -2x = -3x + 15 x = 15 11. 16ˣ⁻² * (4²)ˣ⁺³ = 8 2⁴ˣ⁻⁸ * 2⁴ˣ⁺⁶ = 2³ 8x + 4 = 3 8x = -1 x = -1/8 Simplify the following by pulling out the indicated GCF. 12. 2x⁴ - 9x² - 18x; x x(2x³ - 9x - 18) 13. x⁴ - 9x² - 18x⁴; x⁴ x⁴(1 - 9x² - 18x⁴) Divide the first polynomial by the second using Long Division and again using Synthetic Division. Your final answer should be in the notation of: Q(x) + r(x)/d(x) 14. 25n³ - 20n² - 12n; 5n + 2 5n² - 6n Divide the first polynomial by the second using Long Division. Your final answer should be in the notation of: Q(x) + r(x)/d(x) 15. 25a³ - 8d³ + 14ad² + 5a²d; 5a - 2d 5a² + 3ad + 4d² ____________________ 5a - 2d | 25a³ + 5a²d + 14ad² - 8d³ -25a³ + 10a²d --------------------- 15a²d + 14ad² - 8d³ -15a²d + 6ad² --------------------- 20ad² - 8d³ -20ad² + 8d³ --------------------- 0 16. When a polynomial P(x) is divided by x + 2, the quotient is x² + 3x - 5 and the remainder is -4. Find P(x) P(x) = (x² + 3x - 5)(x + 2) - 4 P(x) = x³ + 5x² + x - 14 17. Find k such that when x³ + 3kx² - 4k²x - 8 is divided by x - 2, the remainder is zero. 2 | 1 3k -4k² -8 | 2 6k+4 -8k²+12k+8 ------------------------ 1 3k+2 -8k²+12k+8 -8k² + 12k + 8 = 0 -4k(2k - 3) = 0 k = 0, 3/2 Divide the first polynomial by the second using the method of your choice. Your final answer should be in the notation of: Q(x) + r(x)/d(x) 18. x³ - 2x² - 3x - 10; x - 3 + i x² + (1 - i)x + (5 - 4i) + (1 - 17i)/(x - 3 + i) Use the Factor Theorem to determine whether the binomial is a factor of the given polynomial. 19. x + 2; P(x) = 3x⁴ - 5x² - 8 P(-2) = 3(-2)⁴ - 5(-2)² - 8 P(-2) = 20 No 20. x + i; P(x) = x⁸ - x⁶ + x⁵ - x² + 1 P(-i) = (-i)⁸ - (-i)⁶ + (-i)⁵ - (-i)² + 1 = (-1)⁴(i)⁴ - (-1)³(i)⁶ + (-1)²(i)⁵ - (-1)²(i)² + 1 = 1 - (-1) - i - 1 + 1 = 2 Solve equation given the indicated root(s). 22. x⁴ - 7x³ + 17x² - 17x + 6 = 0; 1, 2 (Double Root) 1 | 1 -7 17 -17 6 | 1 -6 11 -6 ------------------ 1 -6 11 -6 0 2 | 1 -6 11 -6 | 2 -8 6 ------------------ 1 -4 3 0 (x - 1)(x - 2)²(x - 3) = 0 x = 1, 2, 3 For Questions 23a-d refer to the equation x⁴ + 6x³ + 13x² + 12x + 4 = 0 23a. List all possible rational roots of the polynomial. ±1, ±2, ±4 23b. According to Descartes' Rule, what is the possible number of positive roots? 0 23c. According to Descartes' Rule, what is the possible number of negative roots? 4, 2, or 0 23d. Find all roots of the polynomial. -1 | 1 6 13 12 4 | -1 -5 -8 -4 ------------------ 1 5 8 4 0 -2 | 1 5 8 4 | -2 -6 -4 ------------------ 1 3 2 0 (x + 1)(x + 2)(x² + 3x + 2) = 0 (x + 1)(x + 2)(x + 1)(x + 2) = 0 x = -1, -2 (Both Double Roots) For Questions 24a-d refer to the equation x⁴ - 2x³ - 3x² - 4x + 4 = 0 24a. List all possible rational roots of the polynomial. ±1, ±2, ±4 24b. According to Descartes' Rule, what is the possible number of positive roots? 2 or 0 24c. According to Descartes' Rule, what is the possible number of negative roots? 2 or 0 24d. Find all roots of the polynomial. 1 | 1 -2 -3 -4 4 | 1 -1 -4 -8 ------------------ 1 -1 -4 -8 0 2 | 1 -1 -4 -8 | 2 2 -4 ------------------ 1 1 -2 -4 (x - 1)(x - 2)(x² + 2x + 2) = 0 x² + 2x + 2 = 0 x = -1 ± √(-1) x = 1, 2, ±i√2