SOLUTION: ) Consider the polynomial f(x) = x3 + 6x2 – 25x + 18. Find all of the zeros of the given polynomial. Be sure to show work.

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Question 150023: ) Consider the polynomial f(x) = x3 + 6x2 – 25x + 18.
Find all of the zeros of the given polynomial. Be sure to show work.

Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

First, let's list all of the possible rational zeros.

Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients


So let's list the factors of 18 (the last coefficient):



Now let's list the factors of 1 (the first coefficient):



Now let's divide each factor of the last coefficient by each factor of the first coefficient









Now simplify

These are all the distinct rational zeros of the function that could occur







Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
1|16-2518
| 17-18
17-180

Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
2|16-2518
| 216-18
18-90

Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
3|16-2518
| 3276
19224

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
6|16-2518
| 672282
11247300

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
9|16-2518
| 9135990
1151101008

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
18|16-2518
| 184327326
1244077344

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
-1|16-2518
| -1-530
15-3048

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
-2|16-2518
| -2-866
14-3384

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
-3|16-2518
| -3-9102
13-34120

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
-6|16-2518
| -60150
10-25168

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
-9|16-2518
| -927-18
1-320

Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
-18|16-2518
| -18216-3438
1-12191-3420

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


==================================================

Summary:

So the rational zeros of are

, , or


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