Activity Sheet 12
1 – 4. Describe the end behavior of the graph of each.
1.
f x    x 3  12 x 2  x
2.
f x   x 6  6 x 4  9 x 2
3.
f x   x 5  5 x 3  4 x
4.
f x    x 4  1
5 – 6. Find the zeros for each polynomial function and give the multiplicity of each zero. State
whether the graph crosses or touches the x-axis at each zero. (Hint: On #6, factor by grouping.)
5.
f x   2x  1x  2 x  5
2
3
6.
f x   x 3  5 x 2  25 x  125
7 – 12. For each polynomial function below:
a)
Use the Leading Coefficient Test to determine the graph’s end behavior.
b)
Find the zeros.
c)
State whether the graph crosses or touches the x-axis at each zero.
d)
Find the y-intercept, and then graph the function.
7.
f x   x 3  x 2  9 x  9
8.
f x   4 x  x 3
9.
f x   2 x 3  3x 2  8x  12
10.
f x    x 4  25x 2
11.
f x    x 4  6 x 3  9 x 2
12.
f x   3x 4  15x 3
13.
Use synthetic division to determine whether x  3 is a factor of x 3  7 x 2  5 x  6 .
14.
Use synthetic division to determine whether x  3 is a factor of 2 x 3  8 x 2  x  15 .
55.
Use synthetic division to divide f x   x 3  4 x 2  x  6 by x  1 . Use the result to find all
zeros of f x  .
66.
Use synthetic division to divide f x   x 3  2 x 2  x  2 by x  1 . Use the result to find all
zeros of f x  .
17.
Solve 2 x 3  5 x 2  x  2  0 given that 2 is a zero of f x   2 x 3  5 x 2  x  2 .
18.
Solve 2 x 3  3x 2  11x  6  0 given that  2 is a zero of f x   2 x 3  3x 2  11x  6 .
19 – 22. Use the Rational Zero Theorem to list all possible rational zeros for each function.
19.
f x   x 3  x 2  4 x  4
20.
f x   3x 4  11x 3  x 2  19 x  6
21.
f x   4 x 4  x 3  5 x 2  2 x  6
22.
f x   x 5  x 4  7 x 3  7 x 2  12 x  12
23 – 28. In problems 35 – 40,
a)
List all possible rational zeros of the function.
b)
Use synthetic division to test the possible rational zeros and find an actual zero.
c)
Use the zero from part (b) to find all zeros of the polynomial function.
23.
f x   x 3  x 2  4 x  4
24.
f x   x 3  2 x 2  11x  12
25.
f x   2 x 3  3x 2  11x  6
26.
f x   2 x 3  5 x 2  x  2
27.
f x   3x 3  7 x 2  22 x  8
28.
f x   3x 3  8x 2  15x  4
29 – 36. In problems 35 – 40,
a)
List all possible rational roots
b)
Use synthetic division to test the possible rational roots and find an actual root.
c)
Use the root from part (b) and solve the equation.
29.
x 3  2 x 2  11x  12  0
30.
x3  2x 2  7x  4  0
31.
x 3  10 x  12  0
32.
x 3  5 x 2  17 x  13  0
33.
6 x 3  25 x 2  24 x  5  0
34.
2 x 3  5x 2  6 x  4  0
35.
x 4  2 x 3  5x 2  8x  4  0
36.
x 4  2 x 2  16 x  15  0
37 – 42. Use Descartes’ Rule of Signs to determine the possible number of positive and negative real
zeros for each given function.
37.
f x   x 3  2 x 2  5 x  4
38.
f x   x 3  7 x 2  x  7
39.
f  x   5 x 3  3x 2  3x  1
40.
f x   2 x 3  x 2  x  7
41.
f x   2 x 4  5 x 3  x 2  6 x  4
42.
f x   4 x 4  x 3  5 x 2  2 x  6
43 – 50. Find all zeros of the polynomials or solve the given polynomial equation. Use the Rational Zero
Theorem, Descartes’ Rule of Signs, and possibly the graph of the polynomial function.
43.
f x   x 3  4 x 2  7 x  10
44.
2x3  x 2  9x  4  0
45.
x 4  3x 3  20 x 2  24 x  8  0
46.
f x   3x 4  11x 3  x 2  19 x  6
47.
f x   x 3  x 2  25 x  25
48.
f x   x 3  8 x 2  25x  26
49.
f x   x 4  37 x 2  36
50.
f x   16 x 4  36 x 3  16 x 2  x  30