Question

divide.
\\((-14z^{7}v^{6} - 8z^{4}v^{5} + 10z^{7}v) \div (2z^{4}v^{5})\\)
simplify your answer as much as possible.

Explanation:

Step1: Divide each term by \(2z^4v^5\)

We use the rule of dividing a polynomial by a monomial, which is to divide each term of the polynomial by the monomial. So we have:
\(\frac{-14z^{7}v^{6}}{2z^{4}v^{5}}-\frac{8z^{4}v^{5}}{2z^{4}v^{5}}+\frac{10z^{7}v}{2z^{4}v^{5}}\)

Step2: Simplify each fraction using exponent rules

For the first term: \(\frac{-14}{2}z^{7 - 4}v^{6 - 5}=-7z^{3}v\)
For the second term: \(\frac{-8}{2}z^{4 - 4}v^{5 - 5}=-4z^{0}v^{0}=-4\) (since \(z^{0} = 1\) and \(v^{0}=1\) for \(z
eq0\) and \(v
eq0\))
For the third term: \(\frac{10}{2}z^{7 - 4}v^{1 - 5}=5z^{3}v^{-4}=\frac{5z^{3}}{v^{4}}\) (using the rule \(a^{-n}=\frac{1}{a^{n}}\))

Answer:

\(-7z^{3}v - 4+\frac{5z^{3}}{v^{4}}\)