Question

simplify the expression completely if possible.
\\(\dfrac{4x}{x^3 + 7x^2}\\)

Explanation:

Step1: Factor the denominator

First, we factor the denominator \(x^{3}+7x^{2}\). We can factor out the greatest common factor, which is \(x^{2}\). So, \(x^{3}+7x^{2}=x^{2}(x + 7)\).
The expression now becomes \(\frac{4x}{x^{2}(x + 7)}\).

Step2: Simplify the fraction

We can simplify the fraction by canceling out the common factors in the numerator and the denominator. The numerator has \(x\) and the denominator has \(x^{2}\). Using the rule of exponents \(\frac{x^{a}}{x^{b}}=x^{a - b}\) (for \(x
eq0\)), we have \(\frac{x}{x^{2}}=\frac{1}{x}\) (since \(x^{1-2}=x^{-1}=\frac{1}{x}\)).
So, canceling one \(x\) from the numerator and one \(x\) from the denominator (leaving \(x\) in the denominator), we get \(\frac{4}{x(x + 7)}\) (where \(x
eq0\) and \(x
eq - 7\) to avoid division by zero).

Answer:

\(\frac{4}{x(x + 7)}\) (for \(x
eq0\) and \(x
eq - 7\))