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Chapter 2: Problem 98

Find the value of \(k\) such that \(x-3\) is a factor of \(x^{3}-k x^{2}+2 k x-12\).

Short Answer

Expert verified
The value of \(k\) such that \(x-3\) is a factor of \(x^{3}-k x^{2}+2 k x-12\) is \(k = 5\).

Step by step solution

01

Apply the Factor Theorem

Substitute \(x = 3\) into the equation given by \(x^{3}-k x^{2}+2 k x-12 = 0\). This gives \(3^3 - k \cdot 3^2 + 2k \cdot 3 - 12 = 0\), which simplifies to \(27 - 9k + 6k - 12 = 0\).
02

Simplify the expression

Combine like terms to form a simpler equation in terms of \(k\). This gives \(-3k + 15 = 0\).
03

Solve for \(k\)

Rearrange \(-3k + 15 = 0\) to give \(k = 5\).

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