Selected Answers to Math 390 Review Exam 3I. Polynomials: Perform the indicated operations. A. 1. (3x² - 5x + 4) + (7x² + 9x - 8) = 3x² - 5x + 4 + 7x² + 9x - 8 = 10x² + 4x - 4 2. (7x² - 5x + 3) - (-2x² + 6x - 4) = 7x² - 5x + 3 + 2x² - 6x + 4 = 9x² - 11x + 7 3. 5x(x - 4) - 3x(2x + 1) = 5x² - 20x - 6x² - 3x = -x² - 23x 4. (6x² y)(4x³ y² ) = twenty four x to the fifth power y to the third power. There is no way to indicate an exponent other than 2 or 3 hence the answer in words. 5. (3x - 8)(4x + 1) = 12x² + 3x - 32x - 8 = 12x² - 29x - 8 6. (2x² + 3x - 1)(x² + 5x + 4) = 2x to the fourth power + 10x³ + 8x² + 3x³ + 15x² + 12x - x² - 5x - 4 = 2x to the fourth power + 13³ + 22x² + 7x - 4 7. (4a³ )² = 16 a to the sixth power 8. (5x - 3)² = 25x² - 30x + 9 9. (35x³ y³ )/(7xy² ) = 5x² y 10. (14x³ - 22x)/(2x) = (14x³ )/(2x) - (22x)/2x = 7x² - 11 11. Answer: 2x² + 4x + 9 + 14/(x - 2) B. 1. Subtract x² + 4x - 3 from 2x² + 7x - 8. (2x² + 7x - 8) - (x² + 4x - 3) = 2x² + 7x - 8 - x² - 4x + 3 = x² + 3x - 5 2. What is the degree of the polynomial 4x² + 7x³ - 3x + 9 ? The polynomial is of degree three. II. Factoring. A. Completely factor. 1. 3x² - 9x - 30 = 3(x² - 3x - 10) = 3(x - 5)(x + 2) 2. 6x² - 7x - 10 = (6x + 5 )(x - 2 ) 3. 50x² - 32 = 2(25x² - 16) = 2(5x + 4)(5x - 4) 4. 49x² - 42x + 9 = (7x - 3)² Did you recognize the fact the trinomial was a perfect square trinomial? 5. x² - 5x - 36 = (x - 9)(x + 4) 6. 125x³ - 64 = (5x - 4)(25x² + 20x + 16) 7. 27x³ + 8 = (3x + 2)(9x² - 6x + 4) B. Solve for x. 1. 2x² - 3 = -5x 2x² + 5x - 3 = 0 (2x - 1)(x + 3) = 0 Either 2x - 1 = 0 or x + 3 = 0 Either 2x = 1 or x = -3 Either x = 1/2 or x = -3 2. 4x² - 20x + 25 = 0 (2x - 5)² = 0 2x - 5 = 0 2x = 5 x = 5/2 3. 5x² + 6x - 8 = 0 (5x - 4)(x + 2) = 0 Either 5x - 4 = 0 or x + 2 = 0 Either 5x = 4 or x = -2 Either x = 4/5 or x = -2 C. Stated Problems. 1. If three times the square of a certain integer is added to the integer itself, the sum is 10. What is the integer? Let x denote the integer. 3x² + x = 10 3x&$178 + x - 10 = 0 (3x - 5)(x + 2) = 0 Either 3x - 5 = 0 or x + 2 = 0 3x = 5 or x = -2 x = 5/3 or x = -2 Note 5/3 is not an integer. Hence the integer is -2. 2. The base of a triangle is four feet shorter than twice its altitude. If the area of the triangle is 48 square feet, find its dimensions. Let x denote the altitude. Then 2x - 4 denotes the base. Remeber the area A of a triangle is given by the formula A = 1/2(bh) 48 = 1/2(x)(2x - 4) 96 = x(2x - 4) 96 = 2x² - 4x 0 = 2x² - 4x - 96 0 = 2(x² - 2x - 48) 0 = 2(x - 8)(x + 6) Either x - 8 = 0 or x + 6 = 0 Either x = 8 or x = -6 Since the altitude can't be negative the altitude is 8, the base is 2x - 4 or 2(8) - 4 = 12. Copyright © 1995 - Present. SSmyrl |