Question

divide. if there is a remainder, include it as a simplified fraction. \\((y^3 + 7y^2 + 10y) \div (y + 5)\\)

Explanation:

Step1: Factor the numerator

First, factor the polynomial \(y^{3}+7y^{2}+10y\). We can factor out a common factor of \(y\) first:
\(y^{3}+7y^{2}+10y = y(y^{2}+7y + 10)\)
Then, factor the quadratic \(y^{2}+7y + 10\). We need two numbers that multiply to \(10\) and add to \(7\). Those numbers are \(2\) and \(5\), so:
\(y^{2}+7y + 10=(y + 2)(y + 5)\)
So the numerator becomes \(y(y + 2)(y + 5)\)

Step2: Divide by the denominator

Now we divide \(y(y + 2)(y + 5)\) by \((y + 5)\). The \((y + 5)\) terms cancel out (assuming \(y
eq - 5\)):
\(\frac{y(y + 2)(y + 5)}{y + 5}=y(y + 2)\)

Step3: Expand the result (optional, but to simplify)

Expand \(y(y + 2)\) using the distributive property:
\(y(y + 2)=y^{2}+2y\)

Answer:

\(y^{2}+2y\)