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Pre-Calculus Unit 3 Review Name ______________________ Pd: _______ Complete the square on the following quadratic equations to get them into Standard Vertex Format y = a(x – h)2 + k , then find the vertex point, AOS, and opening direction: 1. y = x2 – 4x + 5 2. y = -4x2 – 24x + 30 Vertex: ___________ Vertex: ________________ AOS: ____________ AOS: _________________ Opening: _________ Opening: ______________ Find the Standard Vertex Equation of the parabola that passes through the given point and Vertex Point: 3. Vertex (1, 1) and Point (3, 5) 4. Vertex (-4, 0) and Point (2, 6) #5 - 8: Are the following polynomial functions in one variable? If yes, state the degree and leading coefficient; if no, tell why not. 5. 4 3 1 1 f ( x) x3 x 2 x 3 2 9 99 6. f ( x) 2 x 3 5 x3 7. f ( x) x 2 3 y 2 8. p( x) 3x 4 3x 2x 7 4x3 4 3 #9 - 10: Determine if the given possible root is in fact a root of the given polynomial equation. Yes or No (Show work for credit!) 9. x = 3; 2 x3 7 x 2 x 3 0 10. 11. Consider the polynomial F(x) = 13x9 4 x 2 2 x10 17 . What is the degree of this polynomial? What is the leading coefficient? How many complex roots does it have? x = -3 ; x 4 5 x 2 36 0 12. If x = 5 is a solution of P(x), state a factor of P(x). 13. If (x + 17) is a factor of P(x), state a root of P(x). 14. If x = 7 is a root of P(x), state an x-intercept of the graph of P(x). 15. If (3,0) is a point of the graph of P(x), state a factor of P(x). 16. For each of the given polynomials, write the end behavior in limit notation, and give the yintercept. a. c. f ( x) 2 x 6 3 x 3 2 x 7 f(x) = (2x + 3)(x – 4)(x + 1) b. p( x) 5x9 6 x8 3x7 2 x 4 3x 2 10 d. p( x) 2 x 4 3x 2 10 17. Sketch each of the following types of polynomial functions. 7th degree; 3 real roots; negative lead coefficient 3rd degree; 2 x-intercepts; positive lead coefficient 18. Write the end behavior of the polynomial in limit notation: y= x(2x + 6)(x + 2)(x - 1) #19 - 21: Solve the polynomial equations over the set of complex numbers. Find the solutions: 19. (x - 3)(x + 4)(7x + 2) = 0 22. Is (x - 3) a factor of (Show all work) 20. x 4 5 x 2 36 0 2x3 - 3x2 - 10x – 9 = 0? 23. Divide the (6x3 - 3x2 + 10x – 5) by (2x + 3) using LONG Division! 21. x3 x 0 24. Given the Linear Factorization of a polynomial below, find all the real zeros and their multiplicity. What does the multiplicity tell you about the graph at each zero? f(x) = (x + 3)(x - 2)2(3x - 2)3(x – 3)4 25. Divide using synthetic division: 3x4 – 4x2 + 6x – 9 divided by (x + 2) 26. Given the following polynomial and one of its factors; find the remaining factors and all zeros. f ( x) 3x3 2 x 2 61x 20; ( x 5) Use the values of A – F in the table below following for questions 27-32: 27. A + C A = 2 + 9i B = 3 – 4i 28. C - D C = -7i + 4 D = -5i 29. AB E=7 F= 30. Conjugate of B 31. Simplify F 32. A B 48 #33 – 34 Sketch the graph of the given polynomial function. Remember, your three keys: xintercepts, y-intercept, and end behavior. You may need to solve the equation first to find the xintercepts, by synthetic division, quadratic formula. 33. f(x) = x3 - 4x 34. y = x3 - 3x2 - 9x + 27 Other review topics on test!! 1. Parent function characteristics, graphing with Transformations 2. Proving inverses by composition 3. Finding inverse functions algebraically 4. Function composition 5. Calculator portion: linear regression