Factoring Polynomial Expressions: A Comprehensive Guide to Solving 2x^4 3x^3 - 4x^2 - 3x 2

Polynomial factorization is a fundamental skill for solving complex equations. This article will guide you through the process of factoring the polynomial expression 2x^4 3x^3 - 4x^2 - 3x 2. We will follow a step-by-step approach, detailing the Rational Root Theorem and the quadratic formula to find all the factors.

Introduction to Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler expressions that, when multiplied together, yield the original polynomial. Understanding this concept is crucial for solving various mathematical problems. In this article, we will use this technique to factor the expression 2x^4 3x^3 - 4x^2 - 3x 2.

Factoring the Expression Step-by-Step

Let's begin by factoring the polynomial expression step-by-step. The given expression is:

2x^4 3x^3 - 4x^2 - 3x 2

Step 1: Rational Root Theorem

The Rational Root Theorem states that any rational root, expressed in its lowest terms, of a polynomial equation anxn an-1xn-1 ... a1x a0 0, is a factor of the constant term a0 divided by a factor of the leading coefficient an. For the polynomial 2x^4 3x^3 - 4x^2 - 3x 2, the constant term is 2 and the leading coefficient is 2. The possible rational roots are thus 1, -1, 2, -2.

One of the possible rational roots is x 1. We can verify this by substituting x 1 into the given polynomial expression:

2(1)^4 3(1)^3 - 4(1)^2 - 3(1) 2 2 3 - 4 - 3 2 0

Hence, x 1 is indeed a root, which means x - 1 is a factor of the polynomial.

We can write the polynomial as:

g(x) f(x) / (x - 1)

Substituting x - 1 into the original polynomial, we get:

g(x) (2x^4 3x^3 - 4x^2 - 3x 2) / (x - 1) 2x^3 5x^2 x - 2

Step 2: Further Factorization

Now, we need to factor the cubic polynomial 2x^3 5x^2 x - 2. For this, we can use the Rational Root Theorem again. The possible rational roots are 1, -1, 2, -2.

Testing x -1, we get:

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

Hence, x -1 is also a root, so x 1 is a factor of the polynomial.

We can write the polynomial as:

h(x) g(x) / (x 1)

Substituting x 1 into the cubic polynomial, we get:

h(x) (2x^3 5x^2 x - 2) / (x 1) 2x^2 3x - 2

Step 3: Factoring the Quadratic

Now, we need to factor the quadratic polynomial 2x^2 3x - 2. We can use the quadratic formula or factor by grouping to find its roots. Using factor by grouping:

2x^2 3x - 2 (2x - 1)(x 2)

So, we can write the entire factorization as:

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

Verification of Factors

To verify that the factors are correct, we substitute the roots back into the original polynomial and check if they result in zero:

p(1) 2(1)^4 3(1)^3 - 4(1)^2 - 3(1) 2 0 p(-1) 2(-1)^4 3(-1)^3 - 4(-1)^2 - 3(-1) 2 0 p(1/2) 2(1/2)^4 3(1/2)^3 - 4(1/2)^2 - 3(1/2) 2 0 p(-2) 2(-2)^4 3(-2)^3 - 4(-2)^2 - 3(-2) 2 0

Since all the values are zero, this confirms that the factors are correct.