Mathematical Investigations III
Name:
Mathematical Investigations III - A View of the World
A Look Back
1. Sketch graphs for the following polynomials, showing important features and labeling significant points (such as x- and y-intercepts).
(a) / (b)(c) / (d)
2. The roots of the polynomial: H(x) = x3 + ax2 + bx + c, are 4, 3, and –2.
(a) Find a and c.
(b) (+) Find b.
3. Let be a polynomial having a bounce point at and a pass-through point at and containing the point . Write an equation for if:
(a) is cubic
(b) is quintic with all real roots
(c) is quintic with two non-real roots
4. Write the equation of a polynomial with roots at 2 and –2, with a double root at –3, and which passes through the point (1,60).
5. Find a cubic polynomial with integer coefficients with two of its zeros being 2/5 and 3 – 2i.
6. Find a cubic polynomial for which two of the roots are and 3 + 2i and whose graph contains the point (1,16).
7. Find an equation of the parabola:
(c) (optional) that contains the points (–1, –10), (1, 4) and (3, –7)
8. Find the vertex and x-intercepts of the parabola .
.
9. Find the vertex and the x-intercepts of the parabola .
10. (a) Find the quotient and the remainder when is divided by x – 2.(c) Find G(2). / (b) Is (x – 2) factor of the polynomial G(x)? Explain.
11. (a) Find the quotient and remainder when the polynomial 2x4 + x3 – 5x2 + 6x – 8 is divided by x2 – 3x – 4. / (b) Is (x2 – 3x – 4) a factor of the polynomial? Explain.
12. (a) Find all ordered pairs of real numbers (x, y) so that (x + 2y) + (3x – 2y)i = 14 + 10i.
(b) Find all ordered pairs of real numbers (x, y) so that x + yi + 2y – xi = 7 + 8i.
(c) Find all pairs of real numbers (x, y) so that (3x – 2yi)(4 + i) + (x + 3yi)(5 – 3i) = 32 – 7i.
13. (a) Find so that (3 – 2i)(x + yi) + (5 – 9i) = 3 + 14i. Give your answer in a+bi form.
(b) Solve for z: (5 + 3i)z – 3(7 + 4i) = 3 + 16i.
14. Simplify the following expressions.
(a) (b)
(c) (5 – 2i)2 (d) 2i7 – 3i17 + 4i30
15. Given the complex number z = 2 + i, find:
(a) (b) |z| (c)
(d) z2 (e) z4 (f) Find all complex solutions of the
equation z4 = –7 + 24i.
16. Simplify the following expressions.
(a) (5 – i)2 / (b) / (c) 3i7 – 2i234
17. Find all zeros, in exact form, of the polynomial f(x) = x4 + 3x3 – 4x2 + 3x + 45.
18. Find all zeros, in exact form, of the polynomial f(x) = 2x4 – 3x3 – 8x2 + 17x + 10.
19. Solve the inequality 2x3 + 7x2 – 20x – 25 ≤ 0
20. Find the equation of a polynomial function of minimal degree producing each graph shown.
(a) (b)
21. (a) Write the equation of a cubic polynomial that will produce the graph at the right.(b) Write the equation of a quintic polynomial with the same bounce points, pass-through points, and y-intercepts as the graph to the right. /
22. Solve for z and make an Argand diagram of the solutions:
(a) z3 + 8i = 0/ (b) z4 + 15z2 – 16 = 0
23. Consider the awful polynomial 2x5 – x4 + 10x3 +305x2 + 398x – 714. Three of its roots are 1, 3, and 3 – 5i. Find the other two (should be very minimal work!).
24. A field with one side bordered by a river is to be enclosed using 400 m of fence. What is the maximum area that can be enclosed? /Poly 14.XXX Rev. S11