Problem 88 - College Algebra

Question

Explain how to simplify a rational expression.

Step-by-Step Solution

Answer

To simplify a rational expression such as \( \frac{15x^2}{20x} \), factorize both the numerator and denominator, identify and cancel out common factors, and compile what's left. Our result in this instance would be \( \frac{3x}{4} \).

Step 1: Factor both the numerator and the denominator

Factoring is a process in which we break down a composite number into its prime factors. Factoring is done in such a way that when the prime factors are multiplied, they give the original number. Use the common techniques such as GCD(Common Divisor), difference of squares, trinomials or by grouping to factorize the expressions. Let's assume the algebraic expression is \( \frac{15x^2}{20x} \). Here, \(15x^2\) can be factored as \(5 \times 3 \times x \times x\) and \(20x\) can be factored as \(5 \times 2 \times 2 \times x\).

Step 2: Cancel out the similar factors in the numerator and the denominator

Now cancel out the similar factors in the numerator and the denominator. In our case, \(5x\) from the numerator can be canceled with \(5x\) from the denominator.

Step 3: Simplify to find the simplified form

After canceling out similar factors, now simplify the remaining factors to write them in form of an algebraic expression. For our example, what we're left with in the numerator is \(3x\) and in the denominator is \(2 \times 2\), or \(4\). So, the simplified form of given rational expression is \( \frac{3x}{4} \).

Step 4: Check the simplified form

Verify if the obtained expression can be simplified any further. If not, then this is the simplest form of the given rational expression. Here, \( \frac{3x}{4} \) is the simplest form. Always remember to check that you cannot further simplify the result - in this case, \( \frac{3x}{4} \) cannot be simplified further, so this is our final answer.