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Mathematical Investigations III Name: Sample Exam - Polynomials No Calculator!! 1. Sketch graph for the following, labeling significant points: 1 a. f ( x) ( x 1) 2 ( x 2)3 2 2. a. b. f ( x) x( x 2)( x 2)2 (3 x) Write a cubic equation that will produce the graph at the right. 4 -2 -8 b. 3. Write all quintic equations that will produce a graph similar to the one at the right. Derive a cubic polynomial with integer coefficients, if two of its zeros are 2/5 and 3 – 2i. Poly 15.1 Rev. S03 Mathematical Investigations III Name: 4. Find the quotient and the remainder when x3 – 5x2 + 6x – 8 is divided by x – 4. Explain why (x – 4) is or is not a factor of the polynomial. 5. Solve for z : (5 3i) z 3(7 4i) 3(3 16i) 6. 1 What is the vertex and the x-intercepts of the parabola, y x 2 2 x 5 5. 3 7. Derive the equation of the parabola whose vertex is (–3, 2) and contains the point (5, –8). 8. Simplify: a. (5 3i) 2 b. Poly 15.2 7 26i 5 2i Rev. S03 Mathematical Investigations III Name: Given the complex equation: z 3 8 0 9. a. Graph the solutions on the complex plane. b. Use what you know about right triangle trigonometry to find the exact roots. 10. Write a polynomial equation for each graph. Poly 15.3 Rev. S03 Mathematical Investigations III Name: 11. Find the sum and product of the roots of Q x 3x4 9 x3 10 x 24 . 12. Write an equation (if possible) for the polynomial function which meets the following criteria: a. quintic, a single root of multiplicity one at x = 5 and no other x-intercepts. b. sixth degree with one root of multiplicity 5 and no other real roots. c. a fourth degree polynomial with a bounce point at x = 2, a pass through points at x 1 and x 3 . Poly 15.4 Rev. S03