Problem 35 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{7 x+35}{x^{2}+5 x} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{7}{x} \).

Step 1: Factor the Numerator

The given expression is \( \frac{7x + 35}{x^2 + 5x} \). First, factor the numerator. Notice that \( 7x + 35 \) can be factored as \( 7(x + 5) \), since both terms have a common factor of 7.

Step 2: Factor the Denominator

Now, let's factor the denominator \( x^2 + 5x \). The expression can be factored by factoring out the common factor \( x \), giving us \( x(x + 5) \).

Step 3: Cancel Common Factors

The expression now looks like \( \frac{7(x + 5)}{x(x + 5)} \). Since \( x + 5 \) is a common factor in both the numerator and the denominator, we can cancel it, provided \( x + 5 eq 0 \). This simplifies the expression to \( \frac{7}{x} \).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler "pieces" that, when multiplied together, give the original expression. It's like finding smaller building blocks that fit together to make something more complex.
For example, in the expression

• \(7x + 35\)

we look for numbers or variables that can be multiplied to give each part of the expression. Notice that both terms, \(7x\) and \(35\), have the number 7 in common.
So, we factor 7 out:

• \(7(x + 5)\)

This process makes the expression simpler and easier to work with. Similarly, when we factor \(x^2 + 5x\), we find that both parts have an \(x\), making it:

• \(x(x + 5)\)

Factoring is especially useful when working with expressions that need simplification, as it helps to identify common factors that can be canceled out.

Simplifying Expressions

Simplifying expressions means rewriting them in a simpler, more digestible form without changing their value or meaning. The goal is to make the expression as straightforward as possible, often by reducing the number of terms.
In the example expression:

• \(\frac{7x + 35}{x^2 + 5x}\)

we factored the numerator and the denominator to become:

• \(\frac{7(x + 5)}{x(x + 5)}\)

Notice that \((x + 5)\) is common in both the top and bottom parts of the fraction. By canceling out these common parts, the expression is simplified to

• \(\frac{7}{x}\)

However, it's important to remember that simplification relies on the condition that the canceled parts are not zero, meaning \(x eq -5\) in this case.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Just like with numerical fractions, you can simplify rational expressions by identifying and canceling common factors.
Consider the original expression

• \(\frac{7x + 35}{x^2 + 5x}\)

This is a rational expression because both the top (7x + 35) and the bottom \((x^2 + 5x)\) are polynomial expressions. Simplifying this involves factoring both parts and reducing common factors, resulting in a simpler form:

• \(\frac{7}{x}\)

One key point while dealing with rational expressions is to be cautious about the values that make the denominator zero, because division by zero is undefined. In this example, \(x eq 0\) and \(x eq -5\) since these values would make the denominator zero. It's crucial to keep these restrictions in mind to correctly interpret the simplified expression.