Question

Write the quotient for \[\left( {3{p^3} - 9{p^2}q - 6p{q^2}} \right) \div \left( { - 3p} \right)\]
A) \[\left( {2{q^2} + 3pq + {p^2}} \right)\]
B) \[\left( {2{q^2} + pq - {p^2}} \right)\]
C) \[\left( {2{q^2} + 3pq - {p^2}} \right)\]
D) None

Answer 1

Hint:
Here we will first write the given expression in the fraction form. Then we will split the fraction form. We will apply the basic division operation and cancel out the similar terms to get the value of the simplified equation.

Complete Step by Step Solution:
The equation is \[\left( {3{p^3} - 9{p^2}q - 6p{q^2}} \right) \div \left( { - 3p} \right)\].
First. we will write the given expression in the fraction form. Therefore, we get
\[ \Rightarrow \left( {3{p^3} - 9{p^2}q - 6p{q^2}} \right) \div \left( { - 3p} \right) = \dfrac{{3{p^3} - 9{p^2}q - 6p{q^2}}}{{ - 3p}}\]
Now we will split the fraction term in the above equation to get it in simplified form. Therefore, we get
\[ \Rightarrow \left( {3{p^3} - 9{p^2}q - 6p{q^2}} \right) \div \left( { - 3p} \right) = \dfrac{{3{p^3}}}{{ - 3p}} - \dfrac{{9{p^2}q}}{{ - 3p}} - \dfrac{{6p{q^2}}}{{ - 3p}}\]
Now we will perform the basic division operation in the above equation to get the simplified form of the equation, we get
\[ \Rightarrow \left( {3{p^3} - 9{p^2}q - 6p{q^2}} \right) \div \left( { - 3p} \right) = - {p^2} + 3pq + 2{q^2}\]
\[ \Rightarrow \left( {3{p^3} - 9{p^2}q - 6p{q^2}} \right) \div \left( { - 3p} \right) = 2{q^2} + 3pq - {p^2}\]
Hence the expression \[\left( {3{p^3} - 9{p^2}q - 6p{q^2}} \right) \div \left( { - 3p} \right)\] is equal to \[\left( {2{q^2} + 3pq - {p^2}} \right)\].

So, option C is the correct option.

Note:
We have expressed the given expression in the form of a fraction. Fraction consists of two parts i.e. numerator and denominator.
There are three different types of fraction:
1) Proper fractions are a fraction having the numerator less, or lower in degree, than the denominator. The value of proper fraction after simplification is always less than 1.
2) An improper Fraction is a fraction where the numerator is greater than or equals the denominator. After the simplification of an improper fraction results in the value which is equal or greater than 1, but not less than 1.
3) A mixed Fraction is the combination of a natural number and fraction. After the simplification of a mixed fraction results in the value which is always greater than 1.