Question

question
simplify: (\frac{20s^{5} + 8s^{4} + 32s^{3}}{-4s^{2}})

Explanation:

Step1: Factor numerator and denominator

First, factor the numerator \(20s^{5}+8s^{4}+32s^{3}\) and the denominator \(-4s^{2}\).
For the numerator, we can factor out the greatest common factor (GCF) of the terms. The GCF of \(20s^{5}\), \(8s^{4}\), and \(32s^{3}\) is \(4s^{3}\).
So, \(20s^{5}+8s^{4}+32s^{3}=4s^{3}(5s^{2} + 2s+8)\)
The denominator is \(-4s^{2}\)

Step2: Simplify the fraction

Now, we have the fraction \(\frac{4s^{3}(5s^{2}+2s + 8)}{-4s^{2}}\)
We can cancel out the common factors. The \(4\) in the numerator and the \(-4\) in the denominator simplify to \(- 1\), and \(s^{3}\) divided by \(s^{2}\) is \(s\) (using the rule of exponents \(\frac{s^{m}}{s^{n}}=s^{m - n}\), here \(m = 3\), \(n=2\), so \(s^{3-2}=s\))
So, \(\frac{4s^{3}(5s^{2}+2s + 8)}{-4s^{2}}=-s(5s^{2}+2s + 8)\)

Step3: Distribute the \(-s\)

Now, distribute \(-s\) into the parentheses:
\(-s\times5s^{2}+(-s)\times2s+(-s)\times8=- 5s^{3}-2s^{2}-8s\)

Answer:

\(-5s^{3}-2s^{2}-8s\)