Factoring Polynomials Using the Factor Theorem and Polynomial Long Division

Understanding and mastering the process of factorizing polynomials is essential for advanced algebra and calculus. In this article, we will guide you through the detailed steps of how to factor the polynomial x^3 - 3x^2 - 9x - 5 using the Factor Theorem and polynomial long division. This article will provide a comprehensive understanding of the techniques involved in factorizing such polynomials.

Finding a Rational Root Using the Factor Theorem

First, we need to find a root of the polynomial f(x). According to the Factor Theorem, if r is a root of f(x), then f(r) 0, and x - r is a factor of f(x).

Step 1: Determine Possible Rational Roots

We start by testing potential rational roots using the Rational Root Theorem. The possible rational roots are derived by taking the factors of the constant term -5 divided by the factors of the leading coefficient 1. Thus, our possible rational roots are ±1, ±5.

Testing Possible Roots

Testing x 1:

Let's evaluate the polynomial at x 1:

f(1) 1^3 - 3(1)^2 - 9(1) - 5 1 - 3 - 9 - 5 -16

1 is not a root.

Testing x -1:

Let's evaluate the polynomial at x -1:

f(-1) (-1)^3 - 3(-1)^2 - 9(-1) - 5 -1 - 3 9 - 5 0

-1 is a root.

Since x -1 is a root, we can conclude that x 1 is a factor of the polynomial.

Using Polynomial Long Division to Factor the Polynomial

Next, we will divide the polynomial x^3 - 3x^2 - 9x - 5 by x 1 using polynomial long division.

Step 2: Polynomial Long Division

Divide the leading term:

frac{x^3}{x} x^2

Multiply and subtract:

x^3 - 3x^2 - 9x - 5 - x^2(x 1) x^3 - 3x^2 - 9x - 5 - x^3 - x^2 -4x^2 - 9x - 5

Divide the leading term again:

frac{-4x^2}{x} -4x

Multiply and subtract:

-4x^2 - 9x - 5 - (-4x)(x 1) -4x^2 - 9x - 5 4x^2 4x -5x - 5

Divide the leading term again:

frac{-5x}{x} -5

Multiply and subtract:

-5x - 5 - (-5)(x 1) -5x - 5 5x 5 0

Since we have a remainder of 0, the division is complete, and we have:

x^3 - 3x^2 - 9x - 5 (x 1)(x^2 - 4x - 5)

Factoring the Quadratic Equation

Step 3: Factor the Quadratic

We next need to factor x^2 - 4x - 5. We look for two numbers that multiply to -5 and add to -4. These numbers are -5 and 1. Thus, we can factor it as:

x^2 - 4x - 5 (x - 5)(x 1)

Final Factorization

Putting it all together, we have:

x^3 - 3x^2 - 9x - 5 (x 1)(x^2 - 4x - 5) (x 1)(x - 5)(x 1) (x 1)^2(x - 5)

Thus, the factorization of x^3 - 3x^2 - 9x - 5 is:

boxed{(x 1)^2(x - 5)}

Understanding these steps will help you factor various polynomials and solve related algebraic equations. For more resources and exercises, consider exploring similar problems or seeking advanced mathematics courses.