Section 3.4 – Zeros of a Polynomial
Find the zeros of f  x    x  2   x  3   x  1  x  4 
2
x20
x  3
2
0
x 1 0
2, -3 (d.r), 1, -4
x40
Find the zeros of f  x   2 x 3  x 2  1 3 x  6
Use your GC to find one of the zeros. Show all the work
that leads to your answer.
6
2
2
1 -13
2
X
4
10
-6
5
-3
0
 x  2 2x2

 5x  3  0
 x  2   2 x  1  x  3   0
x  2,
1
2
,3
Find the zeros of f  x   x 3  1 1x  2 0
Use your GC to find one of the zeros. Show all the work
that leads to your answer.
4
1
0 -11 -20
1
X
4
16
20
4
5
0
x  4x2

 4x  5  0
4 
1 6  4  1  5 
2
4 
x  4,  2  i,  2  i
4
2
 2  i,  2  i
Find the zeros of f  x   x 3  2 x 2  3 x  6
Use your GC to find one of the zeros. Show all the work
that leads to your answer.
6
2
1
-2 -3
1
X
2
0
-6
0
-3
0
x  2x2

3 0
x 3
2
x 
x  2,
3, 
3, 
3
3
Find the zeros of f  x   x 3  5 x 2  2 x  2 4
Use your GC to find one of the zeros. Show all the work
that leads to your answer.
2
1
5
-2 -24
1
X
2
14
24
7
12
0
x  2x2

 7 x  12  0
x  2x  4x  3  0
x  2,  4,  3
No Calculator
Given –2 is a zero of f  x   x 3  2 x 2  5 x  6,
find ALL the zeros of the function.
-2
1
1
-2
-5
6
-2
8
-6
-4
3
0
x  2x2

 4x  3  0
 x  2   x  1  x  3   0
x   2, 1, 3
No Calculator
Given 5 is a zero of f  x   x 4  5 x 3  4 x 2  2 0 x,
find ALL the zeros of the function.
No constant, so 0 is a zero: f  x   x  x 3  5 x 2  4 x  2 0 
5
1
1
-5
-4
20
5
0
-20
0
-4
0


x x  5 x  4  0
2
x x  5x  2x  2  0
x  0, 5, 2,  2
No Calculator
Given -1 and 3 are zeros of f  x   x 4  9 x 3  2 3 x 2  3 x  3 6,
find ALL the zeros of the function.
-1
3
1
1
-9
23
-3
-36
-1
10 -33
36
-10
3
1
-7
33
-21
12
-36
36
0
0
 x  1  x  3   x 2

 7 x  12  0
 x  1  x  3   x  4   x  3   0
x   1, 3  d.r.  , 4
No Calculator
Given
3
2
is a zero of f  x   2 x 3  9 x 2  1 3 x  6,
find ALL the zeros of the function.
3 /2
-9
13
-6
3
-9
6
2
-6
4
0
1
-3
2
2


2x  3  x  3x  2  0
2
 2 x  3   x  1  x  2   0
x 
3
2
, 1, 2
No Calculator
Given
2
3
is a zero of f  x   3 x 3  8 x 2  5 x  6,
find ALL the zeros of the function.
2 /3
-8
-5
6
2
-4
-6
3
-6
-9
0
1
-2
-3
3


3x  2 x  2x  3  0
2
 3 x  2   x  3   x  1  0
x 
2
3
, 3,  1
No Calculator
Given 2 is a zero of f  x   x 3  6 x 2  1 3 x  1 0,
find ALL the zeros of the function.
2
1
1
-6
13
-10
2
-8
10
-4
5
0
x  2x2

 4x  5  0
1 6  4  1  5 
4
2
4
4
2
2  i, 2  i
x  2, 2  i, 2  i
No Calculator
Given –3 is a zero of f  x   x 3  3 x 2  x  3,
find ALL the zeros of the function.
-3
1
1
3
1
3
-3
0
-3
0
1
0
x  3x2

1  0
x  1
2
x  i,  i
x   3, i,  i
No Calculator
Find a polynomial function with real coefficients which has
zeros of 1, -2, and 3.
f  x    x  1  x  2   x  3 

f  x    x  1 x  x  6

2



 6x   x

 x  6
f x  x x  x  6 1 x  x  6

2
f x  x  x
3
2
f  x   x  2x  5x  6
3
2
2
2
No Calculator
Find a polynomial function with real coefficients which has
zeros of 0, 2, -2, and 5.
f x  x x  2x  2x  5


f  x   x  x  x  3 x  10   2  x  3 x  10 


f  x   x  x  3 x  1 0 x    2 x  6 x  2 0 


f  x   x  x  2  x  3 x  10
2
2
2
3
2
2
3
2
f  x   x  x  5 x  4 x  2 0 
f  x   x  5 x  4 x  20 x
4
3
2
No Calculator
Find a polynomial function with real coefficients which has
zeros of 3/2, 2, and 1.
f  x    2 x  3   x  2   x  1

f  x   2x  3  x  3x  2

2



 4 x    3 x
f  x   2x x  3x  2  3 x  3x  2

2
f  x   2x  6x
3
2
f  x   2x  9 x  13 x  6
3
2
2
2

 9x  6

No Calculator
Find a polynomial function with real coefficients which has
zeros of 2 and i.
If i is a root, then –i is a root as well
f  x    x  2   x  i  x  i

f x  x  2 x  1

2




 2
f x  x x  1  2 x  1

2
 
f  x   x  x  2 x
3
2
2
f  x   x  2x  x  2
3
2
No Calculator
Find a polynomial function with real coefficients which has
zeros of 1 and 2 + i.
If 2 + i is a root, then 2 – i is a root as well
f  x    x  1   x  2   i    x  2   i 
f  x    x  1

x  2  i
2
2



f  x    x  1  x  4 x  5 
f  x   x  x  4 x  5   1 x  4 x  5 
f x   x  4x  5x   x  4x  5
f  x    x  1 x  4 x  4  1
2
2
2
3
2
2
f x  x  5x  9x  5
3
2
2