One of the the zeros of the functions F(x)= x^4+2x^3-13x^2-38x-24 is x=-3, find the other zeros of the function.
Use synthetic division with -3, then factor the remainder.
f(x)=2.25 plus 2.75x
To find the other zeros of the function F(x) = x^4 + 2x^3 - 13x^2 - 38x - 24, we can use polynomial long division or synthetic division with the known zero x = -3.
First, let's perform synthetic division to divide F(x) by (x + 3):
-3 | 1 2 -13 -38 -24
| -3 3 30
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1 -1 -10 -8 6
The resulting quotient polynomial is 1x^3 - 1x^2 - 10x - 8 with a remainder of 6.
Now, we have reduced the problem to finding the zeros of this new cubic polynomial: G(x) = x^3 - x^2 - 10x - 8.
To find possible rational zeros, we can use the rational root theorem. According to the theorem, the possible rational zeros of the form p/q, where p is a factor of the constant term (in this case, 8) and q is a factor of the leading coefficient (in this case, 1).
The factors of 8 are ±1, ±2, ±4, ±8.
The factors of 1 are ±1.
So, the possible rational zeros of G(x) are: ±1, ±2, ±4, ±8.
We can now try these possible rational zeros using synthetic division or polynomial long division to see which ones are actual zeros.
Trying x = 1:
1 | 1 -1 -10 -8
| 1 0 -10
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1 0 -10 -18
Trying x = -1:
-1 | 1 -1 -10 -8
| -1 2 8
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1 -2 -8 0
Trying x = 2:
2 | 1 -1 -10 -8
| 2 2 -16
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1 1 -8 -24
Trying x = -2:
-2 | 1 -1 -10 -8
| -2 6 8
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1 -3 -4 0
We can see that x = -2 and x = -1 are zeros of the function G(x).
Therefore, the zeros of the original function F(x) = x^4 + 2x^3 - 13x^2 - 38x - 24 are:
x = -3 (given)
x = -2
x = -1