Question

How do you divide $\dfrac{2{{x}^{2}}+10x+12}{x+3}$ using polynomial long division ?

Answer 1

Hint: We can solve $\dfrac{2{{x}^{2}}+10x+12}{x+3}$ by simple algebraic division. First, we divide the first term of dividend by first term of divisor and the quotient is equal to the first term of the quotient of the total division. Then we will multiply with all the terms of divisor. Then we will subtract the result from $2{{x}^{2}}+10x+12$. Now the result of subtraction is our new dividend and we will continue the same process until the highest power of x in reminder is less than the highest power of x in divisor.

Complete step-by-step solution:
We have to solve $\dfrac{2{{x}^{2}}+10x+12}{x+3}$
First the quotient when we divide $2{{x}^{2}}$ by x is equal to 2x, so the first term of quotient of $\dfrac{2{{x}^{2}}+10x+12}{x+3}$ is equal to 2x
\[x+3\overset{2x}{\overline{\left){2{{x}^{2}}+10x+12}\right.}}\]
Now multiplying 2x with x + 3 we get $2{{x}^{2}}+6x$
\[x+3\overset{2x}{\overline{\left){\begin{align}
  & 2{{x}^{2}}+10x+12 \\
 & \underline{2{{x}^{2}}+6x} \\
\end{align}}\right.}}\]
Now subtracting $2{{x}^{2}}+6x$ form $2{{x}^{2}}+10x+12$ we get 4x + 12
\[x+3\overset{2x}{\overline{\left){\begin{align}
  & 2{{x}^{2}}+10x+12 \\
 & \underline{2{{x}^{2}}+6x} \\
 & 0{{x}^{2}}+4x+12 \\
\end{align}}\right.}}\]
Now diving 4x by x we get 4, so 4 is our next term of quotient, the quotient is updated as 2x + 4. Now multiply 4 with x + 3 we get 4x + 12
Subtracting 4x + 12 from 4x + 12 we get 0.
\[x+3\overset{2x+4}{\overline{\left){\begin{align}
  & 2{{x}^{2}}+10x+12 \\
 & \underline{2{{x}^{2}}+6x} \\
 & \underline{\begin{align}
  & 0{{x}^{2}}+4x+12 \\
 & 0{{x}^{2}}+4x+12 \\
\end{align}} \\
 & 0 \\
\end{align}}\right.}}\]
0 is a constant. We can see the highest power of x in 0 is less than in x + 3
So 0 is the reminder and 2x + 4 is the quotient.
$\dfrac{2{{x}^{2}}+10x+12}{x+3}=2x+4$
We can take 2 common from the quotient
$\Rightarrow \dfrac{2{{x}^{2}}+10x+12}{x+3}=2\left( x+2 \right)$

Note: In the above solution, We can check our answer by factoring the equation $2{{x}^{2}}+10x+12$ By factoring the equation we will get ( x+3) ( 2x+4) . Always remember we should keep the division process continued until the highest power of x in the remainder is less than the highest coefficient of x in the divisor. If we terminate the division before that, then we can still divide the remainder by divisor.