Given that x-1 is a factor of f(x)=x^6-5x^4-10x^2 k, find the value of k. When k has this value, find another factor of f(x) of the form x a, where a is a constant
To find the value of k, we can use the polynomial factor theorem. According to the theorem, if x - c is a factor of a polynomial f(x), then f(c) = 0. In this case, x - 1 is given as a factor of f(x), so we need to find the value of k when f(1) = 0.
Let's substitute x = 1 into the polynomial f(x) and solve for k:
f(x) = x^6 - 5x^4 - 10x^2 + k
f(1) = 1^6 - 5(1^4) - 10(1^2) + k
f(1) = 1 - 5 - 10 + k
f(1) = -14 + k
Since f(1) = 0, we can set -14 + k = 0 and solve for k:
-14 + k = 0
k = 14
Therefore, when k = 14, x - 1 is a factor of f(x).
Now, to find another factor of f(x) of the form x^a, where a is a constant, we can divide f(x) by x - 1.
Using long division or synthetic division, we divide f(x) = x^6 - 5x^4 - 10x^2 + 14 by x - 1:
x^5 + x^4 + 6x^3 + 6x^2 + 4x + 14
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x - 1 | x^6 - 5x^4 - 10x^2 + 0x + 14
x^6 - x^5
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- 4x^5 - 10x^2
- 4x^5 + 4x^4
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- 9x^4 - 10x^2
- 9x^4 + 9x^3
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- x^3 - 10x^2
- x^3 + x^2
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- 11x^2 + 14x
- 11x^2 + 11x
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3x + 14
The remainder is 3x + 14, which means that x - 1 is not a factor of this remainder. So, the other factor of f(x) when k = 14 is x^5 + x^4 + 6x^3 + 6x^2 + 4x + 14.