Problem 41 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{u}{u^{2}+6 u} $$

Step-by-Step Solution

Answer

\(\frac{1}{u+6}\)

Step 1 - Identify common factors

Examine the expression to find any common factors in the numerator and the denominator. The numerator is \(u\) and the denominator is \(u^2 + 6u\).

Step 2 - Factor the denominator

Rewrite the denominator by factoring out the common factor \(u\). So, \(u^2 + 6u = u(u + 6)\).

Step 3 - Simplify the fraction

Now the fraction looks like \frac{u}{u(u+6)}\. There's a common factor \(u\) in both the numerator and the denominator. Cancel out the common factor.

Step 4 - Write the simplified expression

After canceling the common factor \(u\), the simplified fraction is \frac{1}{u+6}\.

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential skill in algebra. It helps us simplify expressions and solve equations more easily. To factor a polynomial, we need to find the greatest common factor (GCF) of its terms. For instance, in the expression given in the exercise, the denominator is a polynomial:

• Denominator: u^2 + 6u

This polynomial has two terms: u^2 and 6u. If we examine these terms, we see that u is a common factor. By factoring out u, we get:

• u^2 + 6u = u(u + 6)

This process makes it possible to see shared elements between the numerator and the denominator. Factoring is crucial because it allows us to simplify complex fractions and make algebraic expressions more manageable.

Simplifying Fractions

Simplifying fractions makes them easier to work with and understand. In algebra, this often involves canceling out common factors between the numerator and the denominator. For our given problem, the fraction has the form: \(\frac{u}{u^2 + 6u}\)After factoring the denominator (as discussed in the previous section), the fraction becomes: \(\frac{u}{u(u + 6)}\)Here, you can see that both the numerator and the denominator have the common factor u. We can cancel out this common factor: \(\frac{u}{u(u + 6)} = \frac{1}{u + 6}\) Always remember to check for common factors and simplify where possible. Simplified fractions are simpler to interpret and further manipulate in solving equations or other algebraic operations.

Common Factors

Identifying common factors is the first step in simplifying algebraic expressions. A common factor is a number or variable that evenly divides both the numerator and the denominator. Let's look at the original fraction from the exercise: \(\frac{u}{u^2 + 6u}\)In this case, the numerator is u, and the denominator is u(u + 6). Clearly, u is a common factor between them. By canceling this common factor, we simplify the expression: \(\frac{u}{u(u + 6)} = \frac{1}{u + 6}\)Always scan your algebraic fractions for common factors. This process makes equations easier to work with and leads to simplified, more efficient solutions.