Question

Divide $p\left( x \right)$ by $g\left( x \right)$ in the following case and verify the division algorithm?
$p\left( x \right)={{x}^{2}}+4x+4$; $g\left( x \right)=x+2$

Answer 1

Hint: Write the polynomial $p\left( x \right)$ in the form ${{a}^{2}}+2ab+{{b}^{2}}$ and use the algebraic identity ${{a}^{2}}+2ab+{{b}^{2}}={{\left( a+b \right)}^{2}}$ to simplify it. Now, divide $p\left( x \right)$ by $g\left( x \right)$ and cancel the common factors to get. If in the denominator we get 1 then the numerator will be the quotient and the remainder will be 0. Finally, verify the division algorithm by using the relation: - dividend = divisor $\times $ quotient + remainder. Solve the R.H.S and if it is equal to the L.H.S then our answers are correct.

Complete step-by-step solution:
Here we have been provided with the polynomial $p\left( x \right)={{x}^{2}}+4x+4$ which is to be divided with the polynomial $g\left( x \right)=x+2$ and we have to verify the division algorithm. First let us see if we can simplify $p\left( x \right)$ or not.
$\Rightarrow p\left( x \right)={{x}^{2}}+2\times 2\times x+{{2}^{2}}$
Clearly this is of the form ${{a}^{2}}+2ab+{{b}^{2}}$ so using the algebraic identity ${{a}^{2}}+2ab+{{b}^{2}}={{\left( a+b \right)}^{2}}$ we can write the polynomial $p\left( x \right)$ as: -
$\Rightarrow p\left( x \right)={{\left( x+2 \right)}^{2}}$
Dividing $p\left( x \right)$ by $g\left( x \right)$ we get,
\[\Rightarrow \dfrac{p\left( x \right)}{g\left( x \right)}=\dfrac{{{\left( x+2 \right)}^{2}}}{\left( x+2 \right)}\]
Cancelling the common factors we get,
\[\Rightarrow \dfrac{p\left( x \right)}{g\left( x \right)}=\dfrac{\left( x+2 \right)}{1}\]
We can see that $g\left( x \right)$ completely divides the polynomial $p\left( x \right)$, so the quotient is $\left( x+2 \right)$ and the remainder is 0.
Now, the division algorithm states that the dividend, divisor, quotient and remainder must verify the relation: - dividend = divisor $\times $ quotient + remainder. So, substituting the values obtained we get,
$\begin{align}
  & \Rightarrow {{\left( x+2 \right)}^{2}}=\left( x+2 \right)\times \left( x+2 \right)+0 \\
 & \Rightarrow {{\left( x+2 \right)}^{2}}={{\left( x+2 \right)}^{2}} \\
\end{align}$
Therefore, we have L.H.S = R.H.S, hence the division algorithm is verified.

Note: Note that if we would not have been able to simplify the dividend polynomial $p\left( x \right)$ then we would have used the long division method to divide $p\left( x \right)$ by $g\left( x \right)$. Here also you can use the long division method. Remember the division algorithm as it is very useful in solving division questions and to find the missing polynomial if three polynomials are given and we have to find the fourth one.