Final answer:
To factor the polynomial x^3 - x^2 - 24x - 36, we would use synthetic division given a factor. Synthetic division involves manipulating the coefficients of the polynomial. Once we find one factor, we can factor further to completely factor the polynomial.
Step-by-step explanation:
To factor the given polynomial function x^3 - x^2 - 24x - 36 using synthetic division, we first need to find a factor of the polynomial. In this case, the question implies that a factor is given, but it is not specified in the question. If the factor is not provided, we would typically use the Rational Root Theorem to test possible factors.
Let's assume that we've found a factor and it is (x - a). To use synthetic division:
Write down the coefficients of the polynomial.
Bring down the leading coefficient.
Multiply a by the number just written down, and write the result underneath the second coefficient.
Add the second coefficient and the number directly below it, and write the result beneath them.
Repeat this process until all coefficients have been used.
If the remainder is zero, then (x - a) is a factor of the polynomial. Once we find one factor, we will be left with a quadratic which can be factored further, either by factoring by grouping, using the quadratic formula, or by additional synthetic division if another linear factor is known.