Question

How do you divide $\dfrac{{3{x^2} + 7x - 20}}{{x + 4}}$ ?

Answer 1

Hint: In this question, we are given a term and we have been asked to divide it and we have to mention the steps as well. You can use a long division method to divide the given term.

Complete step-by-step solution:
We are given a term and we have to divide them. I will use a long division method to divide the given term.
Write $3{x^2} + 7x - 20$ as the dividend and $x + 4$ as the divisor as shown below:
$x + 4)\overline {3{x^2} + 7x - 20} $
Now, next step is to find such a term which when multiplied by $x$ will give us $3{x^2}$ . Such term is $3x\left( { = \dfrac{{3{x^2}}}{x}} \right)$ .
Write this term as the quotient and multiply each term of the divisor with the quotient and write it below the dividend. See below how it can be done:
\[\begin{array}{*{20}{c}}
  {3x} \\
  {x + 4)\overline {3{x^2} + 7x - 20} } \\
  {{\text{ }}3{x^2} + 12x} \\
  {}
\end{array}\]
Now, subtract both the equations like below:
\[\begin{array}{*{20}{c}}
  {3x} \\
  {x + 4)\overline {3{x^2} + 7x - 20} } \\
  {{\text{ }}3{x^2} + 12x} \\
  {\left( - \right)} \\
  {{\text{ }}\overline {{\text{ }} - 5x - 20} }
\end{array}\]
Now, we will have to again find a term with which when $x$ is multiplied, it will give us $ - 5x$. Such term is $ - 5\left( { = \dfrac{{ - 5x}}{x}} \right)$ . Write this term as the quotient and multiply each term of the divisor with the quotient and write it below the previous equation. Now, subtract both the equations. See below how it can be done:
\[\begin{array}{*{20}{c}}
  {{\text{ }}3x - 5} \\
  {x + 4)\overline {3{x^2} + 7x - 20} } \\
  {3{x^2} + 12x} \\
  {\left( - \right)} \\
  {{\text{ }}\overline {{\text{ }} - 5x - 20} } \\
  {{\text{ }} - 5x - 20} \\
  {\left( - \right)} \\
  {{\text{ }}\overline {{\text{ }}0} }
\end{array}\]

Hence, our quotient is $3x - 5$ and our remainder is $0$.

Note: You can also check whether your answer is right or wrong by multiplying the divisor with the quotient and adding the remainder to the product.
Dividend = Divisor $ \times $ Quotient + Remainder
Putting the values in the question,
$3{x^2} + 7x - 20 = \left( {x + 4} \right) \times \left( {3x - 5} \right) + 0$
Opening the brackets to multiply,
$3{x^2} + 7x - 20 = x\left( {3x - 5} \right) + 4\left( {3x - 5} \right)$
Multiplying the terms,
$3{x^2} + 7x - 20 = 3{x^2} - 5x + 12x - 20$
Rearranging the terms,
$3{x^2} + 7x - 20 = 3{x^2} + 7x - 20$
Now, since LHS = RHS, our answer is correct.