Question

How do you long divide $\left( {8{a^2} - 30a + 7} \right) \div \left( {2a - 78} \right)?$

Answer 1

Hint:Here we begin the process of dividing or finding how many times the leftmost digit of the dividend can be divided by the divisor. Then the result or answer which becomes the first digit of the quotient, is multiplied by the divisor and written under the first digit of the dividend. Subtraction is carried out on the first digit of the dividend and the remainder is written. The next digit of the dividend is brought down and then, the process is repeated until all the digits of the dividend are brought down and a remainder is found.

Complete step by step answer:
Step-1:
Write the Given question in long division method,
\[\begin{array}{*{20}{c}}
  {\,\,\,\,\,\,\,\,\,\,\,\,\,4a + 141} \\
  {2a - 78)\overline {8{a^2} - 30a + 7} } \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline {8{a^2} - 312a} \,\,\,\,\,\,\,\,\,\,\,} \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,282a + 7} \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline {282a - 10998} } \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline {11005} }
\end{array}\]
Long division method involving the steps divide, multiply, subtract, then bring the next number down.
Step-1:
Divide: How many times does the divisor fit into the number without remainder? (Use the list of multiplies).Here $8{a^2} \div 2a = 4a$ that is $2a$ goes into $8{a^2}$ in $4a$ times.
Step-2:
Multiply: Multiply the answer to your previous division by the divisor to reach the multiple needed to calculate the remainder (use the list of multiplies). Here $4a$ lots of $2a - 78$ is $8{a^2} - 312a$. This is the term we need to work out the remainder to our first division $8{a^2} \div 2a$
Step-3:
Subtract: Subtract the multiple from the original number to calculate the remainder. Here $(8{a^2} - 30a) - (8{a^2} - 312a) = 282a$ , so this is the remainder to the first division $8{a^2} \div 2a$. This needs to be included in our next step
Step-4:
Bring the next digit down: write down the remainder. Here bringing the $7$ down makes our new term $282a + 7$. Then we will repeat the process again and we get the quotient as well as remainder.

Note:
The polynomial $p(x)$ is the dividend, $g(x)$ is the divisor, $q(x)$ is quotient and $r(x)$ is the remainder. We can write as Dividend=(Divisor$ \times $ Quotient)$ + $ Remainder.
If $r(x)$ is zero, then we say $p(x)$ is a multiple of $g(x)$ . In other words, $g(x)$ divides $p(x)$.
Long division method is set out in a similar way to short the division but uses the memorable step to answer.
The quotient is written above the over bar on top of the dividend.