Math Polynomial and Rational Functions Solution

HPC Midterm Review Name: Key Chapter 2: Polynomial and Rational Functions 1. Express each complex number in standard (a + bi) form. a. √-12 - √-48 = 2i√3 - 4i√3 = -2i√3 b. (√-16 + 3)(√-25 - 2) = (4i + 3)(5i - 2) = 20i² - 8i + 15i - 6 = -20 + 7i - 6 = -26 + 7i c. (3 - 2i)² = (3 - 2i)(3 - 2i) = 9 - 6i - 6i + 4i² = 9 - 12i - 4 = 5 - 12i d. (4 + 3i)(4 - 5i) + (1 + i)² = 16 - 12i + 12i - 15i² + 1 + 2i + i² = 16 + 15 + 1 + 2i - 1 = 21 + 2i e. (10 - 8i)(10 + 8i) = 100 - 64i² = 100 + 64 = 164 f. (1 + i)(1 - i) = 1 - i² = 1 + 1 = 2 g. (3 + 2i)(5 - i) = 15 - 3i + 10i - 2i² = 15 + 7i + 2 = 17 + 7i h. (√3 + 2i)(√3 - 2i) = 3 - 4i² = 3 + 4 = 7 i. i²³ = i³ = -i 2. Solve the following equations for x. a. 12x² + 12 = 25x 12x² - 25x + 12 = 0 (4x - 3)(3x - 4) = 0 x = 3/4, 4/3 b. x² + 4x = 7 x² + 4x - 7 = 0 x = (-4 ± √44) / 2 x = -2 ± √11 c. x² - 6x + 18 = 0 x = (6 ± √36 - 72) / 2 x = (6 ± 6i) / 2 x = 3 ± 3i 3. If P(x) = 4x³ - 5x² + 1, use the remainder theorem (also called synthetic substitution) to find the following: a. P(2) 4 | 4 -5 0 1 | 8 12 24 4 3 12 25 P(2) = 25 b. P(1/2) 1/2 | 4 -5 0 1 | 2 -3/2 3/8 4 -3 -3/2 11/8 P(1/2) = 11/8 c. P(i) i | 4 -5 0 1 | 4i -i -1 4 -1 -i 0 P(i) = 0 4. If -2 is a zero of the polynomial P(x) = 2x³ + kx + 18, find k. 2(-2)³ + k(-2) + 18 = 0 -16 - 2k + 18 = 0 k - 18 = 0 k = 18 5. A polynomial P(x) is divided by (x - 2). The quotient is 2x² + x - 2 and the remainder is -1. Find P(x). P(x) = (2x² + x - 2)(x - 2) + (-1) = 2x³ - 3x² - 4x + 3 6. Two roots of the equation x⁴ + x³ - 5x² + x - 6 = 0 are x = 2 and x = -3. Find all the real roots. [Image of synthetic division] x² + 1 = 0 x = ±i 7. Find all real zeros of f(x) = 2x⁴ + 7x³ - 4x² - 27x - 18. Use the rational root theorem and synthetic division. Possible rational roots: ±1, ±2, ±3, ±6, ±9, ±18 [Image of synthetic division] 2x² + 5x + 3 = 0 (2x + 3)(x + 1) = 0 x = -3/2, -1 8. Without a graphing calculator, sketch the graph of the equation using the x-intercepts, y-intercept and end behavior. f(x) = -x²(x - 1)(x + 2) [Image of the graph of f(x) = -x²(x - 1)(x + 2)] 9. A rectangular enclosure, sub-divided into three congruent pens, is to be made using a barn as one side and 120 meters of fencing for the rest of the enclosure. Find   the value of x that gives the maximum area for the enclosure.   A(x) = x(120 - 4x) = -4x² + 120x x = -b / 2a = -120 / (2 * -4) = 15 meters 10. The path of a diver is approximated by h(x) = -(4/9)x² + (24/9)x + 12, where h(x) is the height (in feet) and x is the horizontal distance (in feet) from the end of the diving board. What is the maximum height of the diver?   y-coordinate of vertex: -b / 2a = -(24/9) / (2 * -4/9) = 3 h(3) = -(4/9)(3)² + (24/9)(3) + 12 = 16 feet 11. Solve for x by factoring. a. 2x³ + 2x² - 4x - 4 = 0 2x²(x + 1) - 4(x + 1) = 0 (2x² - 4)(x + 1) = 0 x = ±√2, -1 b. x⁴ - x³ - 2x² = 0 x²(x² - x - 2) = 0 x²(x - 2)(x + 1) = 0 x = 0, 2, -1 12. Find all real and imaginary roots of the following: a. x² - 4x + 2 = 0 x = (4 ± √(16 - 8)) / 2 x = 2 ± √2 b. 2x⁴ + 3x³ + 7x² - 7x - 5 = 0 Possible rational roots: ±1, ±5, ±1/2, ±5/2 [Image of synthetic division] 2x² + 5x + 5 = 0 x = (-5 ± √(25 - 40)) / 4 x = (-5 ± 2i√5) / 4 13. Find a cubic equation with integral coefficients having the following solutions: a. 3, 2 - √3, and 2 + √3 (x - 3)(x - (2 - √3))(x - (2 + √3)) = 0 (x - 3)((x - 2)² - 3) = 0 (x - 3)(x² - 4x + 1) = 0 x³ - 7x² + 13x - 3 = 0 b. 1 + i and 3 (x - (1 + i))(x - (1 - i))(x - 3) = 0 [(x - 1)² - i²](x - 3) = 0 (x² - 2x + 1 + 1)(x - 3) = 0 x³ - 5x² + 8x - 6 = 0 14. Graph f(x) = { x² if -1 ≤ x ≤ 1 { 1 if x > 1 [Image of the graph of f(x)]