Question

type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar. if ( x + 2 ) is a factor of ( x^3 - 6x^2 - 11x + k ), then ( k = )

Explanation:

Step1: Recall Factor Theorem

If \(x + a\) is a factor of a polynomial \(f(x)\), then \(f(-a)=0\). Here, \(x + 2\) is a factor, so \(a = 2\) and we evaluate \(f(-2)=0\) where \(f(x)=x^{3}-6x^{2}-11x + k\).

Step2: Substitute \(x=-2\) into \(f(x)\)

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Step3: Solve for \(k\)

Since \(f(-2) = 0\) (from Factor Theorem), we set \(-10 + k=0\). Solving for \(k\), we add 10 to both sides: \(k = 10\).

Answer:

10