Question

$\frac{35t^{2}+5t}{8t - t^{4}}div\frac{7t^{2}+22t + 3}{t^{2}+t - 6}$

Explanation:

Step1: Factor the expressions

Factor \(35t^{2}+5t = 5t(7t + 1)\), \(8t - t^{4}=t(8 - t^{3})=t(2 - t)(4 + 2t+t^{2})\), \(7t^{2}+22t + 3=(7t + 1)(t+3)\) and \(t^{2}+t - 6=(t + 3)(t - 2)\)

Step2: Rewrite division as multiplication

\(\frac{35t^{2}+5t}{8t - t^{4}}\div\frac{7t^{2}+22t + 3}{t^{2}+t - 6}=\frac{35t^{2}+5t}{8t - t^{4}}\times\frac{t^{2}+t - 6}{7t^{2}+22t + 3}\)

Step3: Substitute the factored - forms

\(\frac{5t(7t + 1)}{t(2 - t)(4 + 2t+t^{2})}\times\frac{(t + 3)(t - 2)}{(7t + 1)(t + 3)}\)

Step4: Simplify the expression

Cancel out the common factors \((7t + 1)\), \(t\), \((t + 3)\) and note that \(t-2=-(2 - t)\). So we have \(\frac{5\times(-1)}{4 + 2t+t^{2}}=-\frac{5}{t^{2}+2t + 4}\)

Answer:

\(-\frac{5}{t^{2}+2t + 4}\)