Problem 47 - Algebra for College Students

Question

For Problems 9-50, simplify each rational expression. \(\frac{27 x^{4}-x}{6 x^{3}+10 x^{2}-4 x}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{27x^3 - 1}{2(3x^2 + 5x - 2)}\).

Step 1: Factor the Numerator

First, look at the numerator \(27x^4 - x\). Notice that we can factor out an \(x\) to get \(x(27x^3 - 1)\).

Step 2: Factor the Denominator

Now, factor the denominator \(6x^3 + 10x^2 - 4x\). You can factor out \(2x\) from each term, resulting in \(2x(3x^2 + 5x - 2)\).

Step 3: Simplify the Expression

The rational expression after factoring becomes:\[\frac{x(27x^3 - 1)}{2x(3x^2 + 5x - 2)}\]Cancel the common factor \(x\) from the numerator and denominator to simplify it to:\[\frac{27x^3 - 1}{2(3x^2 + 5x - 2)}\]

Step 4: Divide the Remaining Expression

Now that there are no more common factors to cancel, the expression \(\frac{27x^3 - 1}{2(3x^2 + 5x - 2)}\) is in its simplest form.

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down a big problem into smaller, more manageable parts. It's the process of rewriting a polynomial as a product of its factors. This makes it easier to work with, especially when simplifying rational expressions. When factoring, you should always look for a common factor first. This is a number or variable that can be evenly divided into each term in the expression. For instance, in the expression \(27x^4 - x\), you can factor out the common factor \(x\), resulting in \(x(27x^3 - 1)\). This step is crucial because it lays the groundwork for simplifying complex expressions later.

There are several techniques for factoring polynomials:

• Greatest Common Factor (GCF): Always check if there is a common factor you can pull out from each term.
• Factoring by grouping: For polynomials with four or more terms, group terms to make factoring possible.
• Special factorizations: Recognize patterns like difference of squares, perfect square trinomials, and sum/difference of cubes.

Practicing these techniques makes tackling polynomial expressions less intimidating and more approachable for solving or simplifying algebra problems.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing them to their simplest form. This process is similar to simplifying numerical fractions: you look for common factors in the numerator and the denominator and cancel them out.

Let's consider the initial expression given in the problem: \(\frac{27 x^{4}-x}{6 x^{3}+10 x^{2}-4 x}\). First, after factoring the numerator and denominator, we have: \(\frac{x(27x^3 - 1)}{2x(3x^2 + 5x - 2)}\). Here, the \(x\) term can be cancelled out because it's a common factor in both the numerator and the denominator.

This cancellation is similar to how you would simplify the fraction \(\frac{6}{12}\) by cancelling the common factor of 6, reducing it to \(\frac{1}{2}\). After cancelling, what's left in our algebraic example is \(\frac{27x^3 - 1}{2(3x^2 + 5x - 2)}\). Note that this was achieved by just dividing out common factors, which is the essence of simplifying algebraic fractions. Simplifying not only makes expressions look neater but also can make further calculations much easier.

Polynomial Division

Polynomial division is a method used when simplifying rational expressions that can no longer be factored quickly by inspection. It's similar to long division with numbers, but here we deal with variables and coefficients. Once we have simplified our expression to \(\frac{27x^3 - 1}{2(3x^2 + 5x - 2)}\), no further division steps are needed since there are no more common factors between the numerator and the denominator.

Conducting polynomial division involves the following steps:

• Setup: Arrange both the dividend (numerator) and the divisor (denominator) in decreasing order of their powers.
• Divide: Focus on the leading terms of both the dividend and the divisor to create a term of the quotient.
• Multiply and Subtract: Multiply the entire divisor by the new term of the quotient and subtract it from the dividend to find the remainder.
• Repeat: Continue the process until the remainder's degree is less than the divisor's.

In this exercise, it's crucial to understand the need to look for common factors before opting for polynomial division, as it simplifies the process significantly and saves time.