Problem 49 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{-40 x^{3}+24 x^{2}+16 x}{20 x^{3}+28 x^{2}+8 x}\)

Step-by-Step Solution

Answer

\(\frac{2(-5x^2 + 3x + 2)}{5x^2 + 7x + 2}\)

Step 1: Factorize the Numerator

First, look at the numerator \(-40x^3 + 24x^2 + 16x\). We can factor out the greatest common factor, which is \(8x\): \[-40x^3 + 24x^2 + 16x = 8x(-5x^2 + 3x + 2)\]

Step 2: Factorize the Denominator

Now, factor the denominator \(20x^3 + 28x^2 + 8x\). The greatest common factor here is \(4x\): \[20x^3 + 28x^2 + 8x = 4x(5x^2 + 7x + 2)\]

Step 3: Simplify the Rational Expression

Write the expression with the factored terms: \[\frac{8x(-5x^2 + 3x + 2)}{4x(5x^2 + 7x + 2)}\]Cancel out the common factor, \(4x\), from the numerator and denominator:\[\frac{8x}{4x} = 2\]Thus, the expression simplifies to: \[\frac{2(-5x^2 + 3x + 2)}{5x^2 + 7x + 2}\]

Key Concepts

Factoring Polynomials

When you are handling polynomial expressions, factoring is a key skill. It simply means breaking down a larger expression into simpler terms that multiply together to give the original expression. There are various methods for factoring polynomials, but looking for a common factor is often the quickest way.

To factor a polynomial, you should:

• Identify any common factors in each term of the polynomial.
• Then, express the polynomial as the product of these factors and another polynomial.

In the exercise, the first step was to factor the numerator, \(-40x^3 + 24x^2 + 16x\). By spotting a common factor of \(8x\), we could factor it into \(8x(-5x^2 + 3x + 2)\). Meanwhile, the denominator, \(20x^3 + 28x^2 + 8x\), shared a common factor of \(4x\), allowing us to write it as \(4x(5x^2 + 7x + 2)\). Factoring like this makes simplifying the rational expression much easier later on.

Greatest Common Factor

Finding the greatest common factor (GCF) is a fundamental part of simplifying rational expressions. The GCF of terms is the largest factor that divides each term completely. This is crucial because recognizing and using the GCF helps in reducing expressions to their simpler forms.

Steps to identify the GCF in any polynomial include:

• Listing the factors of each coefficient and variable in the terms.
• Finding the largest factor that is common to all terms.

For example, in the original expression with the numerator \(-40x^3 + 24x^2 + 16x\), we determined that \(8x\) was the GCF. Similarly, \(4x\) was the GCF of the denominator, \(20x^3 + 28x^2 + 8x\). By factoring these out, you prepare the polynomial for simplifying the rational expression.

Simplifying Rational Expressions

Simplifying a rational expression involves reducing it to its simplest form by eliminating common factors. Just like you simplify fractional numbers by cancelling common factors, rational expressions are simplified the same way.

Consider these steps when simplifying rational expressions:

• First, factor both the numerator and the denominator completely.
• Cancel out all common factors between the numerator and the denominator.

In our example, after factoring, the expression was \(\frac{8x(-5x^2 + 3x + 2)}{4x(5x^2 + 7x + 2)}\). The common factor \(4x\) from both the numerator and denominator was cancelled, resulting in a simplified expression \(\frac{2(-5x^2 + 3x + 2)}{5x^2 + 7x + 2}\). This form is much neater and easier to evaluate or use further in calculations.