Introduction

This article will guide you through the process of finding the roots of the polynomial equation (x^3 - x^2 - 8x - 12 0). We will explore the Rational Root Theorem, synthetic division, and factorization to determine the solutions accurately.

Using the Rational Root Theorem

The Rational Root Theorem is a powerful tool in algebra that helps us identify potential rational roots of a polynomial equation. For the polynomial (x^3 - x^2 - 8x - 12 0), we start by listing all the potential rational roots, which are the factors of the constant term (-12) divided by the factors of the leading coefficient (1).

Factors of -12

Let's list the factors of -12:

pm 1 pm 2 pm 3 pm 4 pm 6 pm 12

According to the Rational Root Theorem, these factors are the only possible rational roots of the polynomial.

Testing the Possible Roots

Next, we will substitute each of these potential roots into the polynomial to see if they are actual roots.

Testing (x 2)

[2^3 - 2^2 - 8(2) - 12 8 - 4 - 16 - 12 -16 text{not a root}]

Testing (x -2)

[-2^3 - (-2)^2 - 8(-2) - 12 -8 - 4 16 - 12 0 text{is a root}]

Since (x -2) is a root, we can factor the polynomial to find the remaining factors.

Synthetic Division

We now perform synthetic division to factor the polynomial (x^3 - x^2 - 8x - 12) by (x 2).

Synthetic Division: ((x^3 - x^2 - 8x - 12) ÷ (x   2))
  -21-1-8-12
  -26-8
  1-3-20

The result of the synthetic division is the polynomial (x^2 - x - 6).

Factoring the Polynomial

We now factor the resulting quadratic polynomial (x^2 - x - 6).

Factoring (x^2 - x - 6)

The polynomial (x^2 - x - 6) can be factored into ((x - 3)(x 2)).

Complete Factorization

Combining all factors, the complete factorization of the original polynomial is:

Complete Factorization

[x^3 - x^2 - 8x - 12 (x 2)^2(x - 3)]

Thus, the roots of the polynomial (x^3 - x^2 - 8x - 12 0) are:

(x -2) with multiplicity 2 (x 3)

Graphing the Polynomial

For a graphical representation, you can graph the function (y x^3 - x^2 - 8x - 12). This will help you estimate the location of the solutions visually, which is particularly useful when dealing with higher-degree polynomials.

The graph will confirm that the roots are at (x -2) (with a double root) and (x 3).

By combining algebraic methods and graphing, you can confidently find and verify the roots of the polynomial.