Question

question 29 of 42
what is the quotient of the rational expression below?
\\(\frac{x - 4}{3x^3} \div \frac{12x + 1}{x + 3}\\)

a. \\(\frac{x^2 - x - 12}{36x^4 + 3x^3}\\)
b. \\(\frac{x^2 + 7x - 12}{36x^4 + 3x^3}\\)
c. \\(\frac{x^2 - x - 4}{12x^3 + x^2}\\)
d. \\(\frac{x^2 - x - 12}{36x^3 + 1}\\)

Explanation:

Step1: Recall division of rational expressions

To divide two rational expressions, we multiply the first by the reciprocal of the second. So, \(\frac{x - 4}{3x^3}\div\frac{12x + 1}{x + 3}=\frac{x - 4}{3x^3}\times\frac{x + 3}{12x + 1}\)

Step2: Multiply the numerators and denominators

Multiply the numerators: \((x - 4)(x + 3)\). Using the distributive property (FOIL), we get \(x^2+3x - 4x - 12=x^2 - x - 12\). Multiply the denominators: \(3x^3(12x + 1)=3x^3\times12x+3x^3\times1 = 36x^4+3x^3\)

Step3: Form the resulting rational expression

Putting the numerator and denominator together, we have \(\frac{x^2 - x - 12}{36x^4+3x^3}\)

Answer:

A. \(\frac{x^{2}-x - 12}{36x^{4}+3x^{3}}\)