Question

Divide the rational expressions and express in simplest form. When typing your answer for the numerator and denominator be sure to type the term with the variable first.\frac{\left(6p^2+p-12\right)}{\left(8p^2+18p+9\right)}\div \frac{\left(6p^2-11p+4\right)}{\left(2p^2+11p-6\right)}The numerator is AnswerThe denominator is Answer

Answer 1

In order to simplify this expression, first let's put every quadratic polynomial in the factored form:

$\begin{gathered} 6p^2+p-12=6(x+\frac{3}{2})(x-\frac{4}{3}) \\ 8p^2+18p+9=8(x+\frac{3}{2})(x+\frac{3}{4}) \\ 6p^2-11p+4=6(x-\frac{1}{2})(x-\frac{4}{3}) \\ 2p^2+11p-6=2(x+6)(x-\frac{1}{2}) \end{gathered}$

So, switching the division into a multiplication and inverting the second fraction, we have:

$\begin{gathered} \frac{6(x+\frac{3}{2})(x-\frac{4}{3})_{}_{}}{8(x+\frac{3}{2})(x+\frac{3}{4})}\cdot\frac{2(x+6)(x-\frac{1}{2})}{6(x-\frac{1}{2})(x-\frac{4}{3})} \\ =\frac{3(x-\frac{4}{3})}{4(x+\frac{3}{4})}\cdot\frac{(x+6)}{3(x-\frac{4}{3})} \\ =\frac{x+6}{4(x+\frac{3}{4})} \\ =\frac{x+6}{4x+3} \end{gathered}$

Therefore the numerator is x + 6 and the denominator is 4x + 3.