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+(3)/(2))(x-(4)/(3)) \\ 8p^2+18p+9=8(x+(3)/(2))(x+(3)/(4)) \\ 6p^2-11p+4=6(x-(1)/(2))(x-(4)/(3)) \\ 2p^2+11p-6=2(x+6)(x-(1)/(2)) \end{gathered}`

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

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

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