Question

How do you long divide $\left( {2{x^2} + 10x + 12} \right)$ by $x + 3$?

Answer 1

Hint: In this problem we have asked to divide the given quadratic equation by the given term. And we asked to divide the given terms by using the long division method. If we do a long division then divide the first term of the dividend with the first term of the divisor and we write the result as the first term of the quotient. Continuing this process we will get the required solution.

Complete step by step solution:
We asked to divide $\left( {2{x^2} + 10x + 12} \right)$ by $x + 3$

\[\begin{array}{*{20}{c}}
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x + 4} \\
  {x + 3)\overline {2{x^2} + 10x + 12} } \\
  {\,\,\,\,\,\,\,\,\,\underline {\,2{x^2} + 6x\,\,\,\,\,\,\,\,\,\,\,} } \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4x + 12} \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline {4x + 12} } \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0}
\end{array}\]
Considering only the first term of the divisor $x$ and the dividend $2{x^2}$, we see that $2{x^2} \div x = 2x$, so we write $2x$ above $2{x^2}$.
\[\begin{array}{*{20}{c}}
  {2x} \\
  {x + 3)\overline {2{x^2} + 10x + 12} }
\end{array}\]
So $2x$ is the first term of the quotient. Then we multiplied the term $2x$ with the divisor and we wrote the resultant term under the dividend.
Next we multiplied the divisor $x + 3$ by the first term we just wrote above the line to get $2{x^2} + 6x$ which we write under the dividend
\[\begin{array}{*{20}{c}}
  {\begin{array}{*{20}{c}}
  {2x} \\
  {x + 3)\overline {2{x^2} + 10x + 12} }
\end{array}} \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\underline {2{x^2} + 6x\,\,\,\,\,\,\,\,\,\,\,} } \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4x + 12}
\end{array}\]
Multiply the divisor $x + 3$ by the new quotient term $\left( { + 4} \right)$ to get $4x + 12$ then subtract to get a final remainder of $0$. That is,
 \[\begin{array}{*{20}{c}}
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x + 4} \\
  {x + 3)\overline {2{x^2} + 10x + 12} } \\
  {\,\,\,\,\,\,\,\,\,\underline {\,2{x^2} + 6x\,\,\,\,\,\,\,\,\,\,\,} } \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4x + 12} \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline {4x + 12} } \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0}
\end{array}\]

Therefore the remainder is $0$ and the quotient is $2x + 4$.

Note: Long division is a standard division algorithm suitable for dividing multi digit numbers that is simple enough to perform by hands. It breaks down a division problem into a series of easier steps.
First we need to divide the first term of the numerator (i.e. dividend) by the first term of the denominator (i.e. divisor) and put the answer in the place of quotient. Multiply the denominator by that answer and put that below the dividend. Next we have to subtract to create a new polynomial.