Problem 44 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{2 x^{4}-32}{4 x-8}$$

Step-by-Step Solution

Answer

\[\frac{(x^2 + 4)(x + 2)}{2}\]

Step 1: Factor the Numerator

Start by factoring the numerator. The numerator is given as: \[2x^4 - 32\]. Notice that we can factor out a 2 from both terms: \[2(x^4 - 16)\].Now, factor \(x^4 - 16\) as a difference of squares: \[x^4 - 16 = (x^2 + 4)(x^2 - 4)\]. The term \(x^2 - 4\) can be further factored: \[x^2 - 4 = (x + 2)(x - 2)\]. Combining all, the factored numerator is: \[2(x^2 + 4)(x + 2)(x - 2)\].

Step 2: Factor the Denominator

Next, factor the denominator. The denominator is given as: \[4x - 8\]. Notice that we can factor out a 4 from both terms: \[4(x - 2)\].

Step 3: Simplify the Rational Expression

Now, we have the rational expression factored as: \[\frac{2(x^2 + 4)(x + 2)(x - 2)}{4(x - 2)}\]. We can cancel the common term \(x - 2\) from the numerator and the denominator: \[\frac{2(x^2 + 4)(x + 2)}{4}\].

Step 4: Simplify Further

Finally, simplify the coefficients: \[\frac{2}{4} = \frac{1}{2}\]. So, the simplified form of the expression is: \[\frac{(x^2 + 4)(x + 2)}{2}\].

Key Concepts

factoring polynomials

With both the numerator and the denominator factored, simplification becomes easier.

simplifying rational expressions

The result is the simplest version of the original expression. Always cancel before simplifying coefficients.

algebraic fractions

An algebraic fraction, or rational expression, is just a fraction where the numerator and the denominator are polynomials. Simplifying algebraic fractions involves the same principles as simplifying numerical fractions, i.e., factoring and canceling common terms.

It helps to think of polynomials as numbers in these cases. First factorize polynomials completely. Look for common factors and cancel them, simplifying the expression as much as possible.

In our example, the algebraic fraction was initially \(\frac{2x^4 - 32}{4x - 8}\). Through factoring and simplifying, we reduced it to: \(\frac{(x^2 + 4)(x + 2)}{2}\) \This is its lowest term form.