Question

simplify.
\\((-6w^3z^4 - 3w^2z^5 + 4w + 5z) \div (2w^2z)\\)
\\(\circ -3wz^3 - \frac{3z^4}{2} + \frac{2}{wz} + \frac{5}{2w^2}\\)
\\(\circ \frac{1}{-3wz^3} - \frac{3z^4}{2} + \frac{2}{wz} + \frac{5}{2w^2}\\)
\\(\circ -3w^5z^6 - \frac{3w^4z^6}{2} + \frac{2}{w^4z} + \frac{5}{2w^2z^2}\\)
\\(\circ 3wz^3 + \frac{3z^4}{2} + \frac{2}{wz} + \frac{5}{2w^2}\\)

Explanation:

Step1: Divide each term by \(2w^2z\)

We have the expression \((-6w^3z^4 - 3w^2z^5 + 4w + 5z)\div(2w^2z)\), which is equivalent to \(\frac{-6w^3z^4 - 3w^2z^5 + 4w + 5z}{2w^2z}\). We can split this into four separate fractions: \(\frac{-6w^3z^4}{2w^2z}-\frac{3w^2z^5}{2w^2z}+\frac{4w}{2w^2z}+\frac{5z}{2w^2z}\).

Step2: Simplify each fraction

• For \(\frac{-6w^3z^4}{2w^2z}\), we divide the coefficients and subtract the exponents of like bases. The coefficient: \(\frac{-6}{2}=-3\). For \(w\): \(w^{3 - 2}=w\). For \(z\): \(z^{4 - 1}=z^3\). So this simplifies to \(-3wz^3\).
• For \(\frac{-3w^2z^5}{2w^2z}\), the \(w^2\) terms cancel out. The coefficient: \(\frac{-3}{2}\). For \(z\): \(z^{5 - 1}=z^4\). So this simplifies to \(-\frac{3z^4}{2}\).
• For \(\frac{4w}{2w^2z}\), the coefficient: \(\frac{4}{2} = 2\). For \(w\): \(w^{1-2}=w^{-1}=\frac{1}{w}\). So this simplifies to \(\frac{2}{wz}\) (since \(w^{-1}=\frac{1}{w}\), so \(\frac{2}{w z}=\frac{2}{w z}\)).
• For \(\frac{5z}{2w^2z}\), the \(z\) terms cancel out. So this simplifies to \(\frac{5}{2w^2}\).

Putting it all together, we get \(-3wz^3-\frac{3z^4}{2}+\frac{2}{wz}+\frac{5}{2w^2}\).

Answer:

\(-3wz^3 - \frac{3z^4}{2} + \frac{2}{wz} + \frac{5}{2w^2}\) (which corresponds to the first option: \(\boldsymbol{-3wz^3 - \frac{3z^4}{2} + \frac{2}{wz} + \frac{5}{2w^2}}\))