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)
f  x   x3  x  3 x  4

(b) g  x   x 2  x  5  x  3 x  4 
2

(d) k  x   3x3  16 x 2  15 x  18
(c) h  x   2 x2  3x  4  5  x 
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.
Poly 14.1
Rev. S11
Mathematical Investigations III
Name:
3. Let P ( x ) be a polynomial having a bounce point at x  2 and a pass-through point at x  7
and containing the point (0,8) . Write an equation for P ( x ) if:
(a) P ( x ) is cubic
(b) P ( x ) is quintic with all real roots
(c) P ( x ) 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
contains the point (1,16).
Poly 14.2
and 3 + 2i and whose graph
Rev. S11
Mathematical Investigations III
Name:
7. Find an equation of the parabola:
(a) with vertex (–3, 2) and which contains
the point (–1, –10).
(b) with vertex (–3, 2) and which contains
the point (5, –8).
(c) (optional) that contains the points (–1, –10), (1, 4) and (3, –7)
x2
8. Find the vertex and x-intercepts of the parabola y   4 x  2 .
2
.
9. Find the vertex and the x-intercepts of the parabola y  
10. (a) Find the quotient and the remainder when
G  x   3x3  4 x 2  5 x  4 is divided by x – 2.
x2
 2x  5 .
3
(b) Is (x – 2) factor of the polynomial
G(x)? Explain.
(c) Find G(2).
Poly 14.3
Rev. S11
Mathematical Investigations III
Name:
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 x  yi 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.
Poly 14.4
Rev. S11
Mathematical Investigations III
Name:
14. Simplify the following expressions.
(a)
(3  i )(4  i ) (b)
2i
3  4i
(c)
(5 – 2i)2
2i7 – 3i17 + 4i30
15. Given the complex number z = 2 + i, find:
1
(a)
(b) |z|
z
(d) z2
(e) z4
(d)
(c) z
(f) Find all complex solutions of the
equation z4 = –7 + 24i.
16. Simplify the following expressions.
(a) (5 – i)2
(b)
7  26i
5  2i
Poly 14.5
(c) 3i7 – 2i234
Rev. S11
Mathematical Investigations III
Name:
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)
Poly 14.6
Rev. S11
Mathematical Investigations III
Name:
21. (a) Write the equation of a cubic polynomial that
will produce the graph at the right.
4
-2
-8
(b) Write the equation of a quintic polynomial with
the same bounce points, pass-through points, and yintercepts as the graph to the right.
22.
(a)
Solve for z and make an Argand diagram of the solutions:
z3 + 8i = 0
(b) z4 + 15z2 – 16 = 0
imaginary
imaginary
real
real
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 14.7
Rev. S11