Problem 53 - 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

Yes, according to the Factor Theorem, \(x-1\) is a factor of the polynomial \(x^{3}-2x^{2}-11x+12\) because when \(x=1\) is substituted into the polynomial, the result equals to zero.

Step 1: Identifying the Value to Substitute into the Polynomial

The given factor to test is \(x-1\). To find the value of 'x' that must be used, set \(x-1 = 0\), which gives \(x=1\). Hence, \(x=1\) is substituted in the polynomial.

Step 2: Substitution

Substitute \(x=1\) into the polynomial \(x^{3}-2x^{2}-11x+12\). It becomes \(1^{3} - 2(1^{2}) - 11(1) + 12\).

Step 3: Simplifying the Polynomial

Simplify the polynomial. The result becomes \(1 - 2 - 11 + 12\). When simplified further, it equals to zero.

Step 4: Interpreting the Result

Since the simplified result equals zero, it proves that under the Factor Theorem, \(x-1\) is a factor of the polynomial \(x^{3}-2x^{2}-11x+12\).