Question

Work out the following division
\[36\left( x+4 \right)\left( {{x}^{2}}+7x+10 \right)\div 9\left( x+4 \right)\]
(a) \[\left( {{x}^{2}}+7x+10 \right)\]
(b) \[4\left( {{x}^{2}}+7x+6 \right)\]
(c) \[4\left( {{x}^{2}}+7x+10 \right)\]
(d) None of these

Answer 1

Hint: We solve this problem by converting the given divisor that is the polynomial which divides the other polynomial and dividend that is the polynomial which gets divided by other polynomial into a polynomial of certain degree by multiplying all the terms. Then we use the simple division method to get the quotient which is the required answer.

Complete step-by-step answer:
We are asked to solve
\[36\left( x+4 \right)\left( {{x}^{2}}+7x+10 \right)\div 9\left( x+4 \right)\]
Let us assume that the value of above division as \['A'\]
Here, we have the dividend that is the polynomial which gets divided by other polynomial as
\[\Rightarrow P=36\left( x+4 \right)\left( {{x}^{2}}+7x+10 \right)\]
Now, by multiplying the terms in the above equation we get
\[\begin{align}
  & \Rightarrow P=36\left( {{x}^{3}}+7{{x}^{2}}+10x+4{{x}^{2}}+28x+40 \right) \\
 & \Rightarrow P=36{{x}^{3}}+396{{x}^{2}}+1368x+1440 \\
\end{align}\]

Also, we have the divisor that is the polynomial which divides the other polynomial as
\[\begin{align}
  & \Rightarrow Q=9\left( x+4 \right) \\
 & \Rightarrow Q=9x+36 \\
\end{align}\]
Now, we can write the given division as
\[\begin{align}
  & \Rightarrow A=P\div Q \\
 & \Rightarrow A=\left( 36{{x}^{3}}+396{{x}^{2}}+1368x+1440 \right)\div \left( 9x+36 \right) \\
\end{align}\]
Now, let us use the long division method
We know that the division rule that is
\[\text{dividend}=\text{quotient}+\dfrac{\text{remainder}}{\text{divisor}}\]
Now let us try to eliminate the first term in the above equation then we get the quotient and remainder as
\[\Rightarrow A=4{{x}^{2}}+\dfrac{252{{x}^{2}}+1368x+1440}{9x+36}\]
Now by eliminating the first term from the remainder we get the quotient and remainder as
\[\Rightarrow A=4{{x}^{2}}+28x+\dfrac{360x+1440}{9x+36}\]
Now by eliminating the first term from the remainder we get the quotient and remainder as
\[\Rightarrow A=4{{x}^{2}}+28x+40+\dfrac{0}{9x+36}\]
Now, by using the simple division method we get the value of above equation as
\[\Rightarrow A=4{{x}^{2}}+28x+40\]
Now, by taking the common term out from the above equation we get
\[\Rightarrow A=4\left( {{x}^{2}}+7x+10 \right)\]
Therefore the value of given division is
\[\therefore 36\left( x+4 \right)\left( {{x}^{2}}+7x+10 \right)\div 9\left( x+4 \right)=4\left( {{x}^{2}}+7x+10 \right)\]
So, option (c) is the correct answer.

So, the correct answer is “Option (c)”.

Note: Here we have a shortcut method for solving the given problem.
We are given to solve
\[36\left( x+4 \right)\left( {{x}^{2}}+7x+10 \right)\div 9\left( x+4 \right)\]
Let us assume that the value of above division as \['A'\] that is
\[\Rightarrow A=36\left( x+4 \right)\left( {{x}^{2}}+7x+10 \right)\div 9\left( x+4 \right)\]
We know that the from the definition of division that is
\[\Rightarrow a\div b=\dfrac{a}{b}\]
By using this definition to above equation we get
\[\Rightarrow A=\dfrac{36\left( x+4 \right)\left( {{x}^{2}}+7x+10 \right)}{9\left( x+4 \right)}\]
Now, by cancelling the common terms from the numerator and denominator we get
\[\Rightarrow A=4\left( {{x}^{2}}+7x+10 \right)\]
Therefore the value of given division is
\[\therefore 36\left( x+4 \right)\left( {{x}^{2}}+7x+10 \right)\div 9\left( x+4 \right)=4\left( {{x}^{2}}+7x+10 \right)\]
So, option (c) is the correct answer.