Questions: Factor the following polynomial completely. [ (x-12)^2+11(x-12)-242 ] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (x-12)^2+11(x-12)-242= B. The polynomial is prime.

Solution Steps

To factor the given polynomial completely, we can use substitution to simplify the expression. Let \( y = x - 12 \). This transforms the polynomial into a quadratic in terms of \( y \). We can then factor the quadratic expression using standard techniques such as factoring by grouping or using the quadratic formula. Once factored, we substitute back \( x - 12 \) for \( y \) to get the factorization in terms of \( x \).

Let \( y = x - 12 \). The polynomial can be rewritten as: \[ y^2 + 11y - 242 \]

The quadratic \( y^2 + 11y - 242 \) factors to: \[ (y - 23)(y + 10) \]

Substituting back \( y = x - 12 \), we have: \[ (x - 12 - 23)(x - 12 + 10) = (x - 35)(x - 2) \]

Final Answer