Mathematical Investigations III
Name:
Sample Exam - Polynomials
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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