Math 143/7 Some Chapter 3 Review - No calculators bkenny.spring2009
1. Divide the following, writing (a) in the P(x) = D(x)Q(x) + R(x) form,
and (b) in the P(x)
D(x)=Q(x) + R(x)
D(x)form.
a) 5x4−3x2+ 2
x2−3x+ 5 b) x5+ 2
x+ 3
2. Find all the zeros of 3x3−5x2−16x+ 12 = 0, given −2 is a zero.
3. Find all real solutions of
a) 3x5−14x4−14x3+ 36x2+ 43x+ 10 = 0 b) 3x4+ 5x2+ 2 = 0
4. Evaluate the expression, writing the answer in a+bi form.
a) (3 + 5i)(4 −2i) b) 3 + 5i
1−2ic) (√12 −√−3)(3 + √−4) d) √−7√−49
√28
5. Find all solutions (both real and complex) of the equations given.
a) x3−64 = 0 b) x4−1 = 0 c) x6+ 7x3−8 = 0
6. Find the complete factorization of P(x) = x4−2x3−2x2−2x−3.
7. Given each of the following functions, find the intercepts, the asymptotes, and graph
the function.
a) y=4x−8
(x−4)(x+ 1) b) y=3x2+ 6
x2−2x−3
c) y=x2−2x+ 1
x2+ 2x+ 1 d) y=x2−9
2x2+ 1 e) y=3x+ 6
x2+ 2x−8
8. Does there exist a polynomial of degree 6 with integer coefficients that has zeros
i, 2i, 3i, 4i? If so, find it. If not, explain why.
9. What is the remainder when x101 −x4+ 2 is divided by x+ 1?
10. Find the coordinates of all points of intersection of the graphs of y=x4+x2+ 24x
and y= 6x3+ 20. (Do not do this by graphing.)
11. Graph f(x) = x4+x3−2x2, showing the correct x−intercepts and end behavior.
12. Solve, using exact values. x5−x3= 2x
13. Find the degree, the end behavior, and y−intercept of
f(x) = ax8+bx4+cx2+dx +e, a < 0
14. Find bso x+ 2 is factor of f(x)f(x) = x2−bx + 5
15. f(x) = ax2+bx +c
dx3+ex +fhas at most
a) how many x−intercepts? b) how many vertical asymptotes?
16. Graph f(x) = a(x−b)(x−c)2(x−d)2, a, b < 0, c, d > 0, c < d
