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Question 149741: 1. Given the polynomial f(x) = 4x^4 + 5x^3 + 7x^2 - 34x + 8
(a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.
(b) Find all of the zeros of the given polynomial. Show the procedure
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! a)
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 8 (the last coefficient):
Now let's list the factors of 4 (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
b)
Now let's use synthetic division to test each possible zero:
Let's make the synthetic division table for the function  given the possible zero  :
| 1/2 | | | 4 | 5 | 7 | -34 | 8 | | | | | 2 | 7/2 | 21/4 | -115/8 | | | 4 | 7 | 21/2 | -115/4 | -51/8 |
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 make the synthetic division table for the function given the possible zero  :
| 1/4 | | | 4 | 5 | 7 | -34 | 8 | | | | | 1 | 3/2 | 17/8 | -255/32 | | | 4 | 6 | 17/2 | -255/8 | 1/32 |
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 make the synthetic division table for the function given the possible zero  :
| 2 | | | 4 | 5 | 7 | -34 | 8 | | | | | 8 | 26 | 66 | 64 | | | 4 | 13 | 33 | 32 | 72 |
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 make the synthetic division table for the function given the possible zero  :
| 4 | | | 4 | 5 | 7 | -34 | 8 | | | | | 16 | 84 | 364 | 1320 | | | 4 | 21 | 91 | 330 | 1328 |
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 make the synthetic division table for the function given the possible zero  :
| 8 | | | 4 | 5 | 7 | -34 | 8 | | | | | 32 | 296 | 2424 | 19120 | | | 4 | 37 | 303 | 2390 | 19128 |
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 make the synthetic division table for the function given the possible zero  :
| -1 | | | 4 | 5 | 7 | -34 | 8 | | | | | -4 | -1 | -6 | 40 | | | 4 | 1 | 6 | -40 | 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 make the synthetic division table for the function given the possible zero  :
| -1/2 | | | 4 | 5 | 7 | -34 | 8 | | | | | -2 | -3/2 | -11/4 | 147/8 | | | 4 | 3 | 11/2 | -147/4 | 211/8 |
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 make the synthetic division table for the function given the possible zero  :
| -1/4 | | | 4 | 5 | 7 | -34 | 8 | | | | | -1 | -1 | -3/2 | 71/8 | | | 4 | 4 | 6 | -71/2 | 135/8 |
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 make the synthetic division table for the function given the possible zero  :
| -2 | | | 4 | 5 | 7 | -34 | 8 | | | | | -8 | 6 | -26 | 120 | | | 4 | -3 | 13 | -60 | 128 |
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 make the synthetic division table for the function given the possible zero  :
| -4 | | | 4 | 5 | 7 | -34 | 8 | | | | | -16 | 44 | -204 | 952 | | | 4 | -11 | 51 | -238 | 960 |
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 make the synthetic division table for the function given the possible zero  :
| -8 | | | 4 | 5 | 7 | -34 | 8 | | | | | -32 | 216 | -1784 | 14544 | | | 4 | -27 | 223 | -1818 | 14552 |
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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Since none of the possible rational roots are actual roots, this means that the polynomial either has irrational roots or complex roots.
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