Problem 63 - College Algebra

Question

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

Step-by-Step Solution

Answer

By substituting \(x = 1\) into the polynomial and simplifying, we indeed get a zero. Therefore, according to the Factor Theorem, \(x-1\) is a factor of \(x^{3} - 2x^{2} - 11x + 12\).

Step 1: Plug In Value

Plug in \(x = 1\) into the polynomial \(x^{3} - 2x^{2} - 11x + 12\).

Step 2: Simplification

Simplify the expression by calculating the values for \(1^{3} - 2(1^{2}) - 11(1) + 12\).

Step 3: Conclude

If the simplified result is zero, then according to Factor Theorem, \(x-1\) is a factor of the polynomial. If not, then \(x-1\) is not a factor.