Question

Simplify: start fraction, 15, s, to the power 4 , plus, 9, s, cubed, minus, 21, s, squared, minus, 3, s, divided by, minus, 3, s, end fraction
−3s
15s
4
+9s
3
−21s
2
−3s
​

Answer 1

To simplify the expression

\[
\frac{15s^4 + 9s^3 - 21s^2 - 3s}{-3s}
\]

we can factor the numerator.

First, we can factor out a common term from the numerator:

\[
15s^4 + 9s^3 - 21s^2 - 3s
\]

Notice that each term has \(s\) as a common factor:

\[
= s(15s^3 + 9s^2 - 21s - 3)
\]

Next, we can factor \(15s^3 + 9s^2 - 21s - 3\). Pair the terms for easy factoring:

\[
= s((15s^3 + 9s^2) + (-21s - 3))
\]

Factor out the common factors in each pair:

\[
= s(3s^2(5s + 3) - 3(7s + 1))
\]
Now we continue to factor by observing that both groups contain the factor \(3\):

Let's check if we can further factor \(5s + 3\) and \(7s + 1\):

So, now we have:

\[
= s(3(5s + 3)(s - 1))
\]

Therefore, the entire expression can be rewritten as:

\[
= \frac{s(3(5s + 3)(s - 1))}{-3s}
\]

Now we can cancel \(s\) in the numerator and denominator (assuming \(s \neq 0\)):

\[
= \frac{3(5s + 3)(s - 1)}{-3}
\]

The \(3\)s will cancel out:

\[
= -(5s + 3)(s - 1)
\]

Now, expanding this out gives us:

\[
= -[(5s^2 - 5s + 3s - 3)]
\]
\[
= -[5s^2 - 5s + 3s - 3]
\]
\[
= -5s^2 + 2s + 3
\]

Thus, the simplified expression is:

\[
\boxed{-5s^2 + 2s + 3}
\]