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Chapter 3: Problem 63

How can the Factor Theorem be used to determine if \(x-1\) is a factor of \(x^{3}-2 x^{2}-11 x+12 ?\)

Short Answer

Expert verified
Yes, \((x-1)\) is indeed a factor of the polynomial \(x^{3}-2 x^{2}-11 x+12\), as clarified by the Factor Theorem.

Step by step solution

01

Substitution

Substitute \(x=1\) into the polynomial \(x^{3}-2 x^{2}-11 x+12\), such that: \(f(1) = (1)^{3}-2(1)^{2}-11(1)+12\).
02

Simplify

Simplify the expression to verify if \(f(1)\) equals to zero. So, \(f(1) = 1-2-11+12 = 0\).
03

Interpret Result

Since \(f(1) = 0\), by the Factor Theorem, we can conclude that \((x-1)\) is a factor of the given polynomial \(x^{3}-2 x^{2}-11 x+12\).

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