Step-by-Step Guide to Factoring Polynomials: A Comprehensive Approach

Polynomials are a fundamental part of algebra, and understanding how to factor them is crucial for solving equations and simplifying expressions. In this guide, we will walk through a detailed process to factor the polynomial 2x3 - x2 - 12x 9. We will use the Rational Root Theorem, polynomial long division, and the splitting of the middle term technique to achieve this.

Understanding the Rational Root Theorem

The Rational Root Theorem states that any possible rational root of the polynomial 2x3 - x2 - 12x 9 can be found by considering the factors of the constant term (9) divided by the factors of the leading coefficient (2). The potential rational roots are:

±1 ±3 ±9 ±1/2 ±3/2 ±9/2

To find if any of these are actual roots, we test each one by substituting it into the polynomial.

Testing the Potential Rational Root

Let's test x 1 to see if it's a root.

2(1)3 - (1)2 - 12(1) 9 2 - 1 - 12 9 0

Since x 1 results in 0, it is indeed a root of the polynomial. Therefore, x - 1 is a factor.

Polynomial Long Division

Now we will divide the polynomial 2x3 - x2 - 12x 9 by the factor x - 1 using polynomial long division.

Divide the leading term: 2x3 by x, which gives 2x2. Multiply 2x2 by x - 1, resulting in 2x3 - 2x2. Subtract: (2x3 - x2 - 12x 9) - (2x3 - 2x2) 3x2 - 12x 9. Divide the leading term: 3x2 by x, giving 3x. Multiply 3x by x - 1, resulting in 3x2 - 3x. Subtract: (3x2 - 12x 9) - (3x2 - 3x) -9x 9. Divide the leading term: -9x by x, which gives -9. Multiply -9 by x - 1, resulting in -9x 9. Subtract: (-9x 9) - (-9x 9) 0.

The result of the division is 2x2 3x - 9. Therefore, the polynomial can be factored as:

2x3 - x2 - 12x 9 (x - 1)(2x2 3x - 9)

Factoring the Quadratic

The quadratic 2x2 3x - 9 can be factored further. To factor this, we need two numbers that multiply to 2(-9) -18 and add to 3. The numbers 6 and -3 satisfy these conditions.

By splitting the middle term, we can rewrite the quadratic as:

2x2 6x - 3x - 9 2x(x 3) - 3(x 3)

This simplifies to:

(2x - 3)(x 3)

Final Factorization

Combining all the steps, the fully factored form of the polynomial is:

2x3 - x2 - 12x 9 (x - 1)(2x2 3x - 9) (x - 1)(2x - 3)(x 3)

Therefore, the polynomial is factored as:

boxed{(x - 1)(2x - 3)(x 3)}

This method of factoring polynomials is not only systematic but also logical. It involves several techniques such as the Rational Root Theorem, polynomial long division, and the splitting of the middle term. By understanding these concepts, you can effectively factor polynomials that arise in algebra and beyond.