Math Midterm Review 3 Solution Key

Advanced Math Midterm Review Name: Key Chapter 3 Use Synthetic Division to divide the first polynomial by the second. 1. (4x³ + 5x² + 6x - 9) ÷ (x + 2) -2 | 4 5 6 -9 | -8 6 -24 -------------------- 4 -3 12 -33 4x² - 3x + 12 - 33/(x+2) 2. (8x⁴ - 12x³ + 10x² + 5x + 1) ÷ (x - 1/2) 1/2 | 8 -12 10 5 1 | 4 -4 3 4 -------------------- 8 -8 6 8 5 8x³ - 8x² + 6x + 8 + 5/(x - 1/2) 3. (x³ - 5x² + 3) ÷ x x | 1 -5 0 3 | 1 -4 -4 -------------------- 1 -4 -4 -1 x² - 4x - 4 - 1/x 4. (x⁴ - 64) ÷ (x + 4) -4 | 1 0 0 0 -64 | -4 16 -64 256 -------------------- 1 -4 16 -64 192 x³ - 4x² + 16x - 64 + 192/(x + 4) Use Synthetic Division and the Remainder Theorem to find P(c). 5. P(x) = 2x³ - x² + 3x - 1, c = 3 Synthetic Division: 3 | 2 -1 3 -1 | 6 15 54 -------------------- 2 5 18 53 Remainder Theorem: 2(3)³ - (3)² + 3(3) - 1 = 53 6. P(x) = 3x⁴ - 6x² + 7, c = -2 Synthetic Division: -2 | 3 0 -6 0 7 | -6 12 -12 24 -------------------- 3 -6 6 -12 31 Remainder Theorem: 3(-2)⁴ - 6(-2)² + 7 = 31 Use Synthetic Division and the Factor Theorem to determine whether the given binomial is a factor of P(x). 7. P(x) = 3x³ + 4x² - 27x - 36, x + 4 -4 | 3 4 -27 -36 | -12 32 -20 -------------------- 3 -8 5 -56 NOT A FACTOR 8. P(x) = x⁴ - 25x² + 144, x - 3 3 | 1 0 -25 0 144 | 3 9 -48 144 -------------------- 1 3 -16 -48 0 YES, IT'S A FACTOR Use Synthetic Division to show that c is a zero of P. 9. P(x) = 2x³ - 5x² - 4x - 3, c = -1/2 -1/2 | 2 -5 -4 -3 | -1 3 1/2 -------------------- 2 -6 -1 -5/2 REMAINDER EQUALS ZERO 10. P(x) = 2x⁴ - 34x³ + 70x² - 153x + 45, c = 15 15 | 2 -34 70 -153 45 | 30 -240 150 -45 -------------------- 2 -4 10 -3 0 REMAINDER EQUALS ZERO Examine the leading term and the degree of the polynomial to determine the far-left and far-right behavior (end behavior) of the graph of the polynomial function. 11. P(x) = -6x⁴ + 3x² - x - 2 x → ∞, f(x) → -∞ (DOWN TO RIGHT) x → -∞, f(x) → -∞ (DOWN TO LEFT) 12. P(x) = (2/3)x³ - 4x² + 7x - 1 x → ∞, f(x) → ∞ (UP TO RIGHT) x → -∞, f(x) → -∞ (DOWN TO LEFT) 13. P(x) = -3x³ + 4x² + 5x - 6 x → ∞, f(x) → -∞ (DOWN TO RIGHT) x → -∞, f(x) → ∞ (UP TO LEFT) 14. P(x) = x² - 4x + 4 x → ∞, f(x) → ∞ (UP TO RIGHT) x → -∞, f(x) → ∞ (UP TO LEFT) Find the real zeros of the polynomial function by factoring. Identify the multiplicity of each zero. 15. P(x) = x⁴ - 6x³ + 8x² 0 = x²(x² - 6x + 8) 0 = x²(x - 2)(x - 4) x = 0 (mult. 2), x = 2, x = 4 (all mult. 1) 16. P(x) = x⁴ + 13x² + 36 0 = (x² + 9)(x² + 4) x² + 9 = 0 or x² + 4 = 0 x² = -9 or x² = -4 No real solutions (all imaginary) 17. P(x) = -x³ + 2x² + 15x 0 = -x(x² - 2x - 15) 0 = -x(x - 5)(x + 3) x = 0, x = 5, x = -3 (all mult. 1) 18. P(x) = x⁵ - 5x³ + 4x 0 = x(x⁴ - 5x² + 4) 0 = x(x² - 1)(x² - 4) 0 = x(x + 1)(x - 1)(x + 2)(x - 2) x = 0, x = -1, x = 1, x = -2, x = 2 (all mult. 1) Use the Intermediate Value Theorem to verify that P has a zero between a and b. 19. P(x) = 3x³ + 7x² + 3x + 7, a = -3, b = -2 P(-3) = 20, P(-2) = 5 Because the y-values change from positive to negative, there must be a zero between -3 and -2. 20. P(x) = x⁴ - x² - x - 4, a = 1.7, b = 1.8 P(1.7) ≈ -0.2379, P(1.8) ≈ 1.4576 Because the y-values change from negative to positive, there must be a zero between 1.7 and 1.8. Determine the x-intercepts of the graph of P(x). For each x-intercept, use the Even and Odd Powers of x   Theorem to determine whether the graph of P(x) crosses the x-axis (passes through) or intersects but does not cross the x-axis (hits and bounces).   21. P(x) = (2x+1)(x-2)(x+5) x = -1/2 (passes through) x = 2 (passes through) x = -5 (passes through) 22. P(x) = -x⁴ + 4x³ - 3x² 0 = -x²(x² - 4x + 3) 0 = -x²(x - 1)(x - 3) x = 0 (hits and bounces) x = 1 (passes through) x = 3 (passes through) Using x-intercepts and end behavior, sketch the graph of #21 and #22. Do not use a graphing calculator. Use the Rational Zero Theorem to list the possible rational zeros for each polynomial function. Then find ALL the zeros (real and/or imaginary). Don't forget to check for multiplicities!   25. P(x) = x³ - 19x - 30 p: 30 q: 1 p/q: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30 1 | 1 0 -19 -30 | 1 1 -18 -48 -------------------- 1 1 -18 -48 NO -1 | 1 0 -19 -30 | -1 1 18 -------------------- 1 -1 -18 -12 NO 2 | 1 0 -19 -30 | 2 4 -30 -------------------- 1 2 -15 -60 NO -2 | 1 0 -19 -30 | -2 4 30 -------------------- 1 -2 -15 0 YES! x² - 2x - 15 = 0 (x-5)(x+3) = 0 x = 5, x = -3 x = -2, -3, 5 26. P(x) = 2x³ + x² - 25x + 12 p: 12 q: 2 p/q: ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2 1 | 2 1 -25 12 | 2 3 -22 -------------------- 2 3 -22 -10 NO -1 | 2 1 -25 12 | -2 1 24 -------------------- 2 -1 -24 36 NO 2 | 2 1 -25 12 | 4 10 -30 -------------------- 2 5 -15 -18 NO -2 | 2 1 -25 12 | -4 6 38 -------------------- 2 -3 -19 50 NO 1/2 | 2 1 -25 12 | 1 1 -12 -------------------- 2 2 -24 0 YES! 2x² + 7x - 4 = 0 (2x-1)(x+4) = 0 x = 1/2, x = -4 x = -4, 1/2, 3 27. P(x) = x⁴ + x³ + 8x² + 8x + 16 p: 16 q: 1 p/q: ±1, ±2, ±4, ±8, ±16 1 | 1 1 8 8 16 | 1 2 10 18 -------------------- 1 2 10 18 34 NO -1 | 1 1 8 8 16 | -1 0 -8 0 -------------------- 1 0 8 0 16 NO 2 | 1 1 8 8 16 | 2 6 28 72 -------------------- 1 3 14 36 88 NO -2 | 1 1 8 8 16 | -2 2 -20 24 -------------------- 1 -1 10 -12 40 NO CHECK AGAIN! 1 | 1 1 8 8 16 | 1 2 10 18 -------------------- 1 2 10 18 34 NO -1 | 1 1 8 8 16 | -1 0 -8 0 -------------------- 1 0 8 0 16 YES! x² + 8 = 0 x² = -8 x = ±2i x = -2, 2i, -2i 28. P(x) = x⁴ - x³ - 2x² + 2x + 1 p: 1 q: 1 p/q: ±1 1 | 1 -1 -2 2 1 | 1 0 -2 0 -------------------- 1 0 -2 0 1 NO -1 | 1 -1 -2 2 1 | -1 2 0 -2 -------------------- 1 -2 0 2 -1 NO CHECK AGAIN! 1 | 1 -1 -2 2 1 | 1 0 -2 0 -------------------- 1 0 -2 0 1 YES! x² - 2x - 1 = 0 CAN'T BE FACTORED... USE QUAD. FORMULA x = (2 ± √((-2)²-4(1)(-1)))/(2(1)) x = (2 ± √(4+4))/2 = (2 ± √8)/2 = 1 ± √2 x = 1, -1, 1 ± √2 Find a polynomial function of lowest degree with integer coefficients that has the given zeros. 29. -5, 2, 4 P(x) = (x+5)(x-2)(x-4)   P(x) = (x²-3x-10)(x-4) P(x) = x³ - 4x² - 3x² + 12x - 10x + 40 P(x) = x³ - 7x² + 2x + 40 30. -1, 1, -2i, 2i P(x) = (x+1)(x-1)(x+2i)(x-2i) P(x) = (x²-1)(x²+4) P(x) = x⁴ + 4x² - x² - 4 P(x) = x⁴ + 3x² - 4 31. 5, 2i, -2i P(x) = (x-5)(x-2i)(x+2i) P(x) = (x-5)(x²+4) P(x) = x³ - 5x² + 4x - 20 32. 1, -1, 5i, -5i P(x) = (x-1)(x+1)(x-5i)(x+5i) P(x) = (x²-1)(x²+25) P(x) = x⁴ + 25x² - x² - 25 P(x) = x⁴ + 24x² - 25