1. If -3, 2, and 5 are zeros of a function, what is the function? (write the
equation).
(x + 3)(x – 2)(x – 5)
=(x2 + x – 6)(x – 5)
=x3 – 4x2 – 11x + 30
2. If -1, 2, 2, and 3 are zeros of a function, what is the function? (write the
equation).
(x – 2)(x – 2)(x – 3) (x + 1)
=(x2 – 4x – 4)(x – 3)(x + 1)
=(x3 – 3x2 +4)(x – 3)
=x4 – 6x3 + 9x2 + 4x – 12
3. List the possible rational zeros for f(x) = x5 – 2x4 + 3x3 – x2 + 5x – 10.
= ±factors of the constant term (the last term)
±factors of the leading coefficient
= ±1, ±2, ±5, ±10
4. List the possible rational zeros for f(x) = 3x3 – x2 – 4x + 36.
= ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36
±1, ±3
1
2
4
= ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36, ± , ± , ±
3
3
3
For the following, find the rational zeros and list the factors.
5. x3 + 3x2 – x – 3
1
1

-1
1

Rational zeros: 1, -1, -3
3
1
4
-1
3
1
(x + 3)
-1
4
3
-3
-3
3
0
Factors: (x – 1)(x + 1)(x + 3)
6. x3 – 9x2 + 24x – 20
2
1

5
1

Rational zeros: 2, 5
-9
2
-7
5
-2
1
24
-14
10
-10
-20
-20
0
Factors: (x – 2)(x – 2)(x – 5)
(x – 2)
7. 2x3 + x2 – 7x – 6
-1
2

2
2

2
Rational zeros: -1, 2, 
1
-2
-1
4
3
-7
1
-6
6
-6
6
0
3
2
Factors: (x + 1)(x – 2)(2x + 3)
(2x + 3)
8. 9x3 + 18x2 + 11x + 2 *
-1
9

1
3
9


9
Rational zeros: -1, 
18
-9
9
11
-9
2
-3
6
-2
2
-2
0
Factors: (x + 1)(3x + 1)(3x + 2)
(9x + 6)
The factors are: (x + 1)(x +
1
)(3)(3x + 2)
3
= (x + 1)(3x + 1)(3x + 2)
= (x + 1)(x +
2
1
, 
3
3
1
)(9x + 6)
3
Factor out a 3 from the last group
Distribute the 3 to the second factor
For number 9, make a list of everything you know about the function, i.e. end
behavior, how many possible zeros, x-intercept(s), y-intercept, etc.
end behavior: rises to the left, rises to the right
4 possible zeros
x-intercepts: use synthetic division to find -2, -1, 1, 3
y – intercept: (0, 6)
9. x4 – x3 – 7x2 + x + 6
Graph
Your graph doesn’t need to
dip as low as this. Just make
sure you have the end
behavior correct, the
important points plotted
(the zeros and y-intercept),
and the basic shape of the
graph