Use factor theorem to factorize the polynomial x³ - 13x - 12.

To factor the polynomial \(x^3 - 13x - 12\) using the Factor Theorem, we first need to find a root of the polynomial. The Factor Theorem states that if \(f(c) = 0\) for some value \(c\), then \((x - c)\) is a factor of the polynomial.

Finding a Root

Let's test some possible rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (-12) divided by the factors of the leading coefficient (1). The factors of -12 are:

• ±1
• ±2
• ±3
• ±4
• ±6
• ±12

We can test these values in the polynomial:

• For \(x = 1\): \(1^3 - 13(1) - 12 = 1 - 13 - 12 = -24\) (not a root)
• For \(x = -1\): \((-1)^3 - 13(-1) - 12 = -1 + 13 - 12 = 0\) (this is a root)

Factoring the Polynomial

Since \(x = -1\) is a root, we can factor the polynomial as follows:

Using synthetic division to divide \(x^3 - 13x - 12\) by \(x + 1\):

• Write down the coefficients: 1, 0, -13, -12
• Perform synthetic division:

The result of the division is \(x^2 - x - 12\). Now we can factor \(x^2 - x - 12\):

Factoring the Quadratic

We need two numbers that multiply to -12 and add to -1. These numbers are -4 and 3. Thus, we can factor the quadratic as:

\(x^2 - x - 12 = (x - 4)(x + 3)\)

Final Factorization

Putting it all together, we have:

\(x^3 - 13x - 12 = (x + 1)(x - 4)(x + 3)\)

This is the complete factorization of the polynomial.