Asked by d
Is (x-2) a factor of f(x)=x^3-8x^2+14x-4?
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Answered by
GPT-4o mini
To determine if \((x-2)\) is a factor of the polynomial \(f(x) = x^3 - 8x^2 + 14x - 4\), we can use the Factor Theorem. According to the Factor Theorem, \((x - c)\) is a factor of a polynomial \(f(x)\) if and only if \(f(c) = 0\).
In this case, we will evaluate \(f(2)\):
\[
f(2) = (2)^3 - 8(2)^2 + 14(2) - 4
\]
Calculating each term:
\[
f(2) = 8 - 8(4) + 28 - 4
\]
\[
f(2) = 8 - 32 + 28 - 4
\]
\[
f(2) = 8 - 32 + 28 - 4 = 0
\]
Since \(f(2) = 0\), it follows that \((x - 2)\) is indeed a factor of the polynomial \(f(x)\).
In this case, we will evaluate \(f(2)\):
\[
f(2) = (2)^3 - 8(2)^2 + 14(2) - 4
\]
Calculating each term:
\[
f(2) = 8 - 8(4) + 28 - 4
\]
\[
f(2) = 8 - 32 + 28 - 4
\]
\[
f(2) = 8 - 32 + 28 - 4 = 0
\]
Since \(f(2) = 0\), it follows that \((x - 2)\) is indeed a factor of the polynomial \(f(x)\).
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