What is the simplest form of $\frac{x^2+5 x-36}{x^2-16}$ ?
We are asked to simplify the expression \(\frac{x^2+5x-36}{x^2-16}\). To do this, we need to factor both the numerator and the denominator separately. The numerator is a quadratic trinomial \(x^2+5x-36\) and the denominator is a difference of squares \(x^2-16\). Identifying these structures allows us to factor them completely.
First, we factor the numerator \(x^2+5x-36\). We look for two numbers that multiply to \(-36\) and add up to \(5\). These numbers are \(9\) and \(-4\), since \(9 \times (-4) = -36\) and \(9 + (-4) = 5\). Thus, the numerator can be written as \((x+9)(x-4)\). Next, we factor the denominator \(x^2-16\). This is a difference of squares, which follows the pattern \(a^2-b^2 = (a-b)(a+b)\). Here, \(x^2-4^2 = (x-4)(x+4)\).
Now we substitute the factored forms back into the original expression:
\[ \frac{x^2+5x-36}{x^2-16} = \frac{(x+9)(x-4)}{(x+4)(x-4)} \]
Assuming \(x
eq 4\), we can cancel the common factor of \((x-4)\) from both the numerator and the denominator:
\[ \frac{(x+9)\cancel{(x-4)}}{(x+4)\cancel{(x-4)}} = \frac{x+9}{x+4} \]
This is the simplest form of the given expression.
Comparing our result to the given options, we find that the simplified form matches the first option. The result is \(\boxed{\frac{x+9}{x+4}}\).
Simplifying rational expressions is a vital skill for solving equations and understanding functions in algebra. For instance, if you were analyzing the speed of an object where its position is given by a complex ratio, simplifying that ratio can make it much easier to predict the object's future position. This technique is used in various fields like physics for calculating rates of change, or in engineering when optimizing dimensions of a structure, where complex mathematical models often boil down to simpler, manageable fractions.