Math Polynomial and Rational Functions

Chapter 2: Polynomial and Rational Functions 1. Convert the quadratic f(x) = -x^2 - 10x - 29 to vertex form, then identify the vertex, any intercepts, and sketch its graph. a. Vertex Form: y + 29 = -(x^2 + 10x + 25) y + 29 = -(x + 5)^2 y = -(x + 5)^2 - 4 b. Vertex: (-5, -4) c. x-intercepts: 0 = -(x + 5)^2 - 4 4 = -(x + 5)^2 -4 = (x + 5)^2 (no real solutions) d. y-intercept: (0, -29) 2. 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) = -4x^2 + 120x To find the maximum, we find the vertex of the parabola: x = -b / 2a = -120 / (2 * -4) = 15 3. The path of a diver is approximated by h(x) = -(4/9)x^2 + (24/9)x + 12 where h(x) is the height (in feet) and x is the horizontal distance (in feet) from the end of a diving board. What is the maximum height of the diver?   To find the maximum height, we find the vertex of the parabola: x = -b / 2a = -(24/9) / (2 * -4/9) = 3 Maximum height: h(3) = -(4/9)(3)^2 + (24/9)(3) + 12 = 16 ft 4. Without a graphing calculator, sketch the graph of the equation. a. f(x) = -x^2(x - 1)(x + 2) b. f(x) = x^4 - 2x^3 - 7x^2 - 4x 5. Given f(x) = x^4 + x^3 - 5x^2 + x - 6, complete the following: a. Find the zeros of f(x). Zeros: -3, 2, ±i b. Write the complete factorization of f(x). f(x) = (x + 3)(x - 2)(x^2 + 1) 6. If -2 is a zero of the polynomial P(x) = 2x^3 + x + k, find k. k = 18 7. Solve the equation: a. 12x^2 + 12 = 25x 12x^2 - 25x + 12 = 0 (4x - 3)(3x - 4) = 0 x = 3/4, 4/3 b. x^4 - 4x^2 + 6x - 4 = 0 (x - 2)(x^2 + 2) = 0 x = 2, ±i√2 c. x^6 + x^4 + 18 = 0 (x^2 + 3)(x^4 - 2x^2 + 6) = 0 x = ±i√3 d. x^4 - x^3 - 2x^2 = 0 x^2(x - 2)(x + 1) = 0 x = 0, 2, -1 e. x^2 + 4x = 7 (x + 2)^2 = 11 x = -2 ± √11 f. 2x^3 + 2x^2 - 4x - 4 = 0 (x + 1)(2x^2 - 2) = 0 x = -1, ±√2 g. 2x^4 + 3x^3 + 7x^2 - 7x - 5 = 0 (x - 1)(x + 1)(2x^2 + 4x + 5) = 0 x = -1, 1, -1 ± i√2 h. 2x^4 + 7x^3 - 4x^2 - 27x - 18 = 0 (x + 3)(2x + 3)(x^2 - 2x - 2) = 0 x = -3, -3/2, 1 ± √3 8. Find a polynomial in expanded form that satisfies the given conditions. a. Zeros: -1, 5 - √3; Degree: 3 (x+1)(x-(5-√3))(x-(5+√3)) (x+1)(x²-10x+22) x³-9x²+12x+22 b. Zeros: 1 ± i and 3; Degree: 4 (x-(1+i))(x-(1-i))(x-3)² (x²-2x+2)(x²-6x+9) x⁴-8x³+23x²-30x+18 9. Simplify. Express each complex number in standard (a+bi) form. a. (√-16 + 3)(√-25 - 2) (4i+3)(5i-2) 20i²-8i+15i-6 -26+7i b. (3-2i)² (3-2i)(3-2i) 9-6i-6i+4i² 5-12i c. 4(3+4i) - 5i(1+i) 12+16i-5i-5i² 17+11i d. (√3 + 2i) / (√3 - 2i) (√3 + 2i)(√3 + 2i) / (√3 - 2i)(√3 + 2i) (3 + 4i + 2√3i) / (3 + 4) (-1 + 4√3i) / 7 e. i⁻²³ + i⁻⁵ 1/i¹³ + 1/i⁵ i/i² + i³/i³ i + (-i) = 0 10. Sketch the rational function by hand. Identify any asymptotes, intercepts, and holes in the graph. a. f(x) = (x²-5x+6) / (x-2) y-intercept: (0, -3) x-intercept(s): (3, 0) horizontal asymptote: none vertical asymptote(s): none hole(s) in the graph: (2, -1) slant asymptote: y = x - 3 10. (continued) Sketch the rational function by hand. Identify any asymptotes, intercepts, and holes in the graph. b. f(x) = 2x² / (x² - 4) Note: This is an even function! y-intercept: (0, 0) (multiplicity 2) x-intercept(s): (0, 0) (multiplicity 2) horizontal asymptote: y = 2 slant asymptote: none vertical asymptote(s): x = ±2 hole(s) in the graph: none c. f(x) = (x² - x + 1) / (x - 3) y-intercept: (0, -1/3) x-intercept(s): none horizontal asymptote: none slant asymptote: y = x + 2 vertical asymptote(s): x = 3 hole(s) in the graph: none