Question

Using remainder theorem, find the remainder when ${x^3} - a{x^2} + 2x - a$divisible by$x - a$.
$A$) $a$
$B$) $a - 2$
$C$) $a - 1$
$D$) $a - 2$

Answer 1

Hint: Here we have to find out the remainder of the given expression by using the remainder theorem. So, we need to apply the remainder theorem first and by doing some simplification we get the required answer.

Complete step-by-step solution:
Given: we have a polynomial written below: ${x^3} - a{x^2} + 2x - a.........(1)$
Now we will apply the remainder theorem: If a polynomial $P(x)$ is divided by the binomial $x - a$, the remainder obtained is $P(a)$.
Substitute $x = a$ in the given polynomial. After that here we apply remainder theorem to get the remainder.
We need to put the given equation equal to zero
Hence we are doing
$ \Rightarrow x - a = 0$
By putting the equation equal to zero we get the root which is
$ \Rightarrow x = a$
Therefore we got the root.
Now putting the value of $x = a$ in the given polynomial equation
$ \Rightarrow {a^3} - a{(a)^2} + 2a - a$
By using the bodmas we will solve firstly brackets
$ \Rightarrow {a^3} - {a^3} + 2a - a$
We can cancel the opposite terms such as ${a^3}$, so we done
$ \Rightarrow 2a - a$
Now we are subtracting the above term as they have same degree
So we get,
$ \Rightarrow a$
The remainder is $a$

Thus the correct option is ($A$) that is $a$.

Note: We could also have found the remainder by the long division method which is a more intuitive method. But this method is mathematically simpler than the long division method. For the long division method, we perform a division in the traditional manner. Just, instead of numbers, we will have polynomials.
 Alternate Method: Long division method
Steps to do long division method:
\[\begin{array}{*{20}{c}}
  {\,\,\,\,\,\,{x^2} + 2} \\
  {x - a)\overline {{x^3} - a{x^2} + 2x - a} } \\
  {\underline { - {x^3} + a{x^2}\,\,\,\,\,} } \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x - a} \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { - 2x + 2a} } \\
  {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a}
\end{array}\]
$1.$ Make sure the polynomial is written in descending order. If any term is missing then use the zero in the place of that term.
$2.$ Divide the terms with the highest power inside the division symbol by the highest power outside the division symbol.
$3.$ Multiply the answer obtained in the previous step by the polynomial in front of the division symbol.
$4.$ Subtract and bring down the next term.
$5.$ Repeat steps $2,3$and $4$until there are no more terms to bring down
$6.$ Write the final answer. The terms remaining after the subtract step is the remainder and must be written as a fraction in the final answer.