Simplify the expression.
$\frac{4 x+36}{x^2+9 x}$
Select the correct choice below and fill in any answer boxes in your choice.
A. $\frac{4 x+36}{x^2+9 x}=$ $\square$ (Simplify your answer.)
B. The expression cannot be simplified.
Answer (1)
Factor the numerator: 4 x + 36 = 4 ( x + 9 ) .
Factor the denominator: x 2 + 9 x = x ( x + 9 ) .
Rewrite the expression: x ( x + 9 ) 4 ( x + 9 ) .
Cancel the common factor ( x + 9 ) : x 4 .
Therefore, the simplified expression is x 4 .
Explanation
Analyze the problem We are asked to simplify the expression x 2 + 9 x 4 x + 36 . This is a rational expression, so we will try to factor both the numerator and the denominator to see if there are any common factors that can be cancelled.
Factor the numerator First, we factor the numerator: 4 x + 36 = 4 ( x + 9 ) .
Factor the denominator Next, we factor the denominator: x 2 + 9 x = x ( x + 9 ) .
Rewrite the expression Now we rewrite the expression with the factored numerator and denominator: x 2 + 9 x 4 x + 36 = x ( x + 9 ) 4 ( x + 9 ) .
Cancel common factors We can cancel the common factor of ( x + 9 ) from the numerator and the denominator, provided that x = − 9 . Also, x cannot be 0 since it is in the denominator. Thus, we have: x ( x + 9 ) 4 ( x + 9 ) = x 4 .
State the simplified expression Therefore, the simplified expression is x 4 .
Examples
Simplifying rational expressions is useful in many areas, such as physics and engineering, where complex formulas can be made easier to work with. For example, in circuit analysis, simplifying expressions can help in calculating the overall resistance or impedance of a circuit. In physics, simplifying expressions can help in understanding the relationship between different physical quantities. This skill is also fundamental for more advanced math courses like calculus, where simplifying expressions is often a necessary step in solving more complex problems.
