Question

find the quotient.
\frac{-30a^{4}b^{3}-30a^{3}b^{4}-30ab}{6ab}
enter the correct answer.

Explanation:

Step1: Split the fraction

We can split the given fraction into three separate fractions:
$$\frac{-30a^{4}b^{3}}{6ab} + \frac{-30a^{3}b^{4}}{6ab} + \frac{-30ab}{6ab}$$

Step2: Simplify each fraction

For the first fraction:

• Divide the coefficients: $\frac{-30}{6} = -5$
• For the variables with $a$: $\frac{a^{4}}{a} = a^{4 - 1} = a^{3}$ (using the rule $\frac{a^{m}}{a^{n}} = a^{m - n}$)
• For the variables with $b$: $\frac{b^{3}}{b} = b^{3 - 1} = b^{2}$

So, the first fraction simplifies to $-5a^{3}b^{2}$

For the second fraction:

• Divide the coefficients: $\frac{-30}{6} = -5$
• For the variables with $a$: $\frac{a^{3}}{a} = a^{3 - 1} = a^{2}$
• For the variables with $b$: $\frac{b^{4}}{b} = b^{4 - 1} = b^{3}$

So, the second fraction simplifies to $-5a^{2}b^{3}$

For the third fraction:

• Divide the coefficients: $\frac{-30}{6} = -5$
• For the variables with $a$: $\frac{a}{a} = 1$
• For the variables with $b$: $\frac{b}{b} = 1$

So, the third fraction simplifies to $-5$

Step3: Combine the simplified fractions

Now, we combine the three simplified fractions:
$$-5a^{3}b^{2} - 5a^{2}b^{3} - 5$$

Answer:

$-5a^{3}b^{2} - 5a^{2}b^{3} - 5$