University:
College Entrance Preparation
Course:
ACT Preparation (Mathematics)
Academic year:

2021
Views:

310

Pages:

5
Author:

Arielle Armstrong

Name: Honors Pre-Calc Hour 14/15 Review Chapter 4 Test Complete the following WITHOUT a calculator. 1. Use the Remainder Theorem to find the remainder when the polynomial 16x^5 - 32x^4 - 81x + 162 is divided by x - 2. State whether the binomial is a factor of the polynomial. Explain your answer. Remainder: 0 Yes or No: Yes Explanation: Remainder = 0, so x - 2 is a factor (Factor Theorem) 21 | 16 -32 0 -81 162 16 -16 -16 96 162 16 -16 -16 15 162 At most how many real solutions would 16x^5 - 32x^4 - 81x + 162 have? 5 At least how many real solutions would 16x^5 - 32x^4 - 81x + 162 have? 1 The left and right hand behavior of 16x^5 - 32x^4 - 81x + 162 is: 2. Use Synthetic Division to divide x^4 - 20x^3 - 14x^2 + 8x + 1 by x + 1. Solution: x^3 - 21x^2 + 7x - 1 + 7/(x + 1) Find f(-1) for the function in problem two. f(-1) = 7 Is x = -1 a root of f(x)? Yes or No: No Explain answer: Remainder is not 0, so x = -1 is not a factor. 3. Use Long Division to find the other factor if x^2 - 8 is a factor of x^3 + 3x^2 - 8x - 24. Other factor: x + 3 x^3 + 3x^2 - 8x - 24 | x^2 - 8 x^3 + 0x^2 - 8x 3x^2 + 0x - 24 3x^2 + 0x - 24 0 4. Solve 3x^2 - 24x = -30 by Completing the Square. Solution: x = 4 ± √6 3x^2 - 24x = -30 x^2 - 8x = -10 x^2 - 8x + 16 = -10 + 16 (x - 4)^2 = 6 x - 4 = ±√6 x = 4 ± √6 5. Given the following roots of f(x), build the factors of f(x), then use the factors to build a polynomial f(x) of the least degree. Roots of f(x): x = 5, x = -3, x = 2 Factors of f(x): (x-5)(x+3)(x-2) Polynomial f(x): (x-5)(x+3)(x-2) = x³-4x²-7x+30 6. Given the following roots of f(x), build the factors of f(x), then use the factors to build a polynomial f(x) of the least degree. Roots of f(x): x = 5i, x = -√3 Factors of f(x): (x-5i)(x+5i)(x+√3)(x-√3) or f(x): (x²-25i²)(x²-3) Polynomial f(x): (x²+25)(x²-3) = x⁴+22x²+75 7. Find the discriminant of 2x²-4x+9=0 and describe the nature of the roots. Value of discriminant: -56 Nature of the roots: 2 non-real conjugates. D = b²-4ac = (-4)²-4(2)(9) = -56 8. Solve -3x²+4=0 by using the Quadratic Formula. Solution: x = ±2√3/3 x = (-b±√(b²-4ac))/2a = (-4±√(16-4(-3)(4)))/2(-3) = ±√12/-6 = ±√2/-1 = ±2√2/2 9. List the possible rational roots of 2x³+17x²+23x-42=0 Solution: ±1, ±2, ±3, ±6, ±7, ±14, ±21, ±42, ±1/2, ±3/2, ±7/2, ±21/2 p: ±1, ±2, ±3, ±6, ±7, ±14, ±21, ±42 q: ±1, ±2 p/q: ±1, ±2, ±3, ±6, ±7, ±14, ±21, ±42, ±1/2, ±3/2, ±7/2, ±21/2 10. Find the number of possible negative real zeros for x⁴-5x³+2x²-12x+6 Solution: 1 or 3