Question

perform the following operation and express in simplest form.\\(\dfrac{x^2 - 36}{3x + 18} \div \dfrac{x^2 + x - 42}{3x^2}\\)

Explanation:

Step1: Factor all polynomials

Factor \(x^2 - 36\) as a difference of squares: \(x^2 - 36=(x + 6)(x - 6)\)
Factor \(3x + 18\) by taking out the GCF: \(3x + 18 = 3(x + 6)\)
Factor \(x^2 + x - 42\) by finding two numbers that multiply to \(-42\) and add to \(1\): \(x^2 + x - 42=(x + 7)(x - 6)\)

Step2: Rewrite division as multiplication by reciprocal

Dividing by a fraction is the same as multiplying by its reciprocal:
\(\frac{(x + 6)(x - 6)}{3(x + 6)}\div\frac{(x + 7)(x - 6)}{3x^2}=\frac{(x + 6)(x - 6)}{3(x + 6)}\times\frac{3x^2}{(x + 7)(x - 6)}\)

Step3: Cancel common factors

Cancel \((x + 6)\), \((x - 6)\), and \(3\) from numerator and denominator:
After canceling, we have \(\frac{x^2}{x + 7}\)

Answer:

\(\boldsymbol{\frac{x^2}{x + 7}}\)