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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 :
Since the remainder  (the right most entry in the last row) is equal to zero, this means that is a zero of
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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 :
Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of
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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 :
Since the remainder  (the right most entry in the last row) is not equal to zero, this means that is not a zero of
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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 | | | 1 | 6 | -25 | 18 | | | | | 6 | 72 | 282 | | | 1 | 12 | 47 | 300 |
Since the remainder  (the right most entry in the last row) is not equal to zero, this means that is not a zero of
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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 | | | 1 | 6 | -25 | 18 | | | | | 9 | 135 | 990 | | | 1 | 15 | 110 | 1008 |
Since the remainder  (the right most entry in the last row) is not equal to zero, this means that is not a zero of
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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 | | | 1 | 6 | -25 | 18 | | | | | 18 | 432 | 7326 | | | 1 | 24 | 407 | 7344 |
Since the remainder  (the right most entry in the last row) is not equal to zero, this means that is not a zero of
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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 | | | 1 | 6 | -25 | 18 | | | | | -1 | -5 | 30 | | | 1 | 5 | -30 | 48 |
Since the remainder  (the right most entry in the last row) is not equal to zero, this means that is not a zero of
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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 | | | 1 | 6 | -25 | 18 | | | | | -2 | -8 | 66 | | | 1 | 4 | -33 | 84 |
Since the remainder  (the right most entry in the last row) is not equal to zero, this means that is not a zero of
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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 | | | 1 | 6 | -25 | 18 | | | | | -3 | -9 | 102 | | | 1 | 3 | -34 | 120 |
Since the remainder  (the right most entry in the last row) is not equal to zero, this means that is not a zero of
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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 | | | 1 | 6 | -25 | 18 | | | | | -6 | 0 | 150 | | | 1 | 0 | -25 | 168 |
Since the remainder  (the right most entry in the last row) is not equal to zero, this means that is not a zero of
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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 :
Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of
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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 | | | 1 | 6 | -25 | 18 | | | | | -18 | 216 | -3438 | | | 1 | -12 | 191 | -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
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Summary:
So the rational zeros of are
 ,  , or 
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