University:
Cornell University
Course:
MATH 1110 | Calculus I
Academic year:

2023
Views:

276

Pages:

5
Author:

Mckayla Mclaughlin

Algebra 2 CP MIDTERM REVIEW Name: Key CHAPTER 4 - QUADRATICS Add or subtract the following polynomials. (Distribute if necessary, and then combine like terms) (2x - 2y) + (5x + 2y) = 7x (5x - 7) - (7x - 6) = -2x - 1 (-3x^3 + 12x^2 - 7x) - (9x^2 + 2x - 7) = -3x^3 + 3x^2 - 9x + 7 Multiply polynomials. Use distribution or FOIL -5x^2y^4(7x^5y + 3x^2y - x^6y^4) = -35x^7y^5 - 15x^4y^7 - 5x^8y^5 (x + 2)(x - 3) = x^2 - x - 6 (4x + 5y)(2x - 7y) = 8x^2 - 18xy - 35y^2 (x + 3)^2 = x^2 + 6x + 9 (x - 4)(x^2 - 4x + 1) = x^3 - 8x^2 + 17x - 4 2(x - 3)(x - 4) = 2x^2 - 14x + 24 True or false? (a + b)^3 = a^3 + b^3 False (a - b)(a^2 + ab + b^2) = a^3 - b^3 True (a + b)(a^2 - ab + b^2) = a^3 + b^3 True Factor the following polynomials. Look for the greatest common factor, difference of squares, trinomials, and sum/difference of cubes 13. x^2 - 3x - 10 = (x - 5)(x + 2) 2y^2 - 50 = 2(y - 5)(y + 5) 3x^2 - 18x - 24 = 3(x - 6)(x + 1) 16. 4a^2 + a - 3 = (4a - 3)(a + 1) 17. x^3 + 3x^2 - 70x = x(x^2 + 3x - 70) = x(x + 10)(x - 7) 18. m^3 - 1 = (m - 1)(m^2 + m + 1) 19. 27r^4y^9 + 18r^3y^6 - 72r^3y^3 = y^3(3ry^6 + 2r^2y^3 - 8r) 20. 81x^4 - 16 = (9x^2 - 4)(9x^2 + 4) = (3x - 2)(3x + 2)(9x^2 + 4) 21. 45x^2 - 80y^2 = 5(9x^2 - 16y^2) = 5(3x - 4y)(3x + 4y) 22. 2a^3 - 8a^3 - 27 = (2a - 3)(4a^2 + 6a + 9) 23. c^2 + 6c - 7 = (c + 7)(c - 1) 24. 6x^2 - 3x - 18 = 3(2x^2 - x - 6) = 3(2x + 3)(x - 2) Solve by factoring 25. x^2 - 2x = 15 = x^2 - 2x - 15 = 0 = (x - 5)(x + 3) = 0 = x = 5, -3 26. 4c^2 = 20c = 4c^2 - 20c = 0 = 4c(c - 5) = 0 = c = 0, 5 27. 3x^2 + 4x - 15 = 0 = 3x^2 + 9x - 5x - 15 = 0 = 3x(x + 3) - 5(x + 3) = 0 = (3x - 5)(x + 3) = 0 = x = 5/3, -3 Simplify 28. (7 - 8i) + (-12 - 4i) = -5 - 12i 29. (10 - 4i) - (7 - 3i) = 3 - i 30. (-3 + 5i) + (18 - 7i) = 15 - 2i 31. (2 + i)(3 - 5i) = 6 - 10i + 3i - 5i^2 = 6 - 7i + 5 = 11 - 7i 32. (7 - 6i)(2 - 3i) = 14 - 21i - 12i + 18i^2 = 14 - 33i - 18 = -4 - 33i 33. (3 + 4i)(3 - 4i) = 9 - 12i + 12i - 16i^2 = 9 + 16 = 25 34. (8 - 6i) / 3i = (8 - 6i) * (3i / 3i) = 24i - 18i^2 / 9i^2 = 18 + 24i / -9 = -2 - 8/3i 35. 3i / (4 + 2i) = 3i * (4 - 2i) / (4 + 2i)(4 - 2i) = 12i - 6i^2 / 16 - 4i^2 = 6 + 12i / 20 = 3/10 + 3/5i Fill in the blanks to complete the square. 36. x^2 - 14x + 49 = (x - 7)^2 37. x^2 + 9x + 81/4 = (x + 9/2)^2 38. m^2 - m + 1/4 = (m - 1/2)^2 Solve by completing the square. 39. x^2 + 2x = 15 = x^2 + 2x + 1 = 15 + 1 = (x + 1)^2 = 16 = x + 1 = ±4 = x = 3, -1 40. 4x^2 - 8x - 12 = 0 = 4(x^2 - 2x - 3) = 0 = x^2 - 2x - 3 = 0 = x^2 - 2x + 1 = 3 + 1 = (x - 1)^2 = 4 = x - 1 = ±2 = x = 3, -1 41. x^2 - 7x + 5 = 0 = x^2 - 7x + 49/4 = 5 + 49/4 = (x - 7/2)^2 = 29/4 = x - 7/2 = ±√29/2 = x = (7 ± √29) / 2 Write each quadratic function in vertex form. Identify the vertex, axis of symmetry, and direction of opening. 42. y = 2x^2 + 16x + 29 = y - 29 = 2(x^2 + 8x) = y + 3 = 2(x + 4)^2 = y = 2(x + 4)^2 - 3 Vertex: (-4, -3) Axis of symmetry: x = -4 Opens up 43. y = 2x^2 + 9x + 2 = y - 2 = 2(x^2 + 9/2x) = y + 81/8 = 2(x + 9/4)^2 = y = 2(x + 9/4)^2 - 81/8 Vertex: (-9/4, -81/8) Axis of symmetry: x = -9/4 Opens up

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