Question

Find the remainder when ${x^{100}}$ is divided by ${x^2} - 3x + 2$.

Answer 1

Hint: Clearly if you do this question by long division, it would consume a lot of time. So you can use the basic division formula $dividend = divisor \times quotient + remainder$ and will evaluate the remainder by putting values of x in the expression such that $divisor \times quotient$ =0

Complete step-by-step answer:
We have,$f(x) = {x^{100}}$ and divide it by $q(x) = {x^2} - 3x + 2$ and evaluate the remainder.
So, the degree of q(x) is 2 and hence the degree of remainder will be less than 2.
Let us assume that remainder $r(x) = Ax + B$
Now we know that in division $dividend = divisor \times quotient + remainder$
So, $f(x) = q(x) \times p(x) + r(x)$ ,where p(x) is quotient.
$ \Rightarrow {x^{100}} = ({x^2} - 3x + 2) \times p(x) + Ax + B$ (1)
Also we can write
 $
  q(x) = {x^2} - 3x + 2 \\
   = {x^2} - x - 2x + 2 \\
   = x(x - 1) - 2(x - 1) \\
   = (x - 1)(x - 2) \\
$
We can see that zeros of q(x) are 1 and 2 .So we will put x=1 and x=2 in (1) s that quotient p(x) can be removed from the expression.
Now we have ${x^{100}} = p(x) \times (x - 1)(x - 2) + Ax + B$ (2)
Putting x=1 in (2), we get
  $
  {1^{100}} = p(x) \times (1 - 1)(1 - 2) + A(1) + B \\
   \Rightarrow 1 = 0 + A + B \\
   \Rightarrow A + B = 1{\text{ }}(3) \\
$
And x=2 in (2), we get
 $
  {2^{100}} = p(x) \times (x - 1)(2 - 2) + A(2) + B \\
   \Rightarrow {2^{100}} = 0 + 2A + B \\
   \Rightarrow 2A + B = {2^{100}}{\text{ (4)}} \\
$
You can see in (3) and (4) that we have two unknowns A and B and two equations. So we can solve it.
Now subtracting (3) from (4)
$\begin{array}{*{20}{c}}
  {}&{2A}& + & B & = &{}&{{2^{100}}} \\
  {}& A & + & B & = &{}&1 \\
   - &{}& - &{}&{}& - &{} \\
\hline
  {}&A&{}&{}& = &{}&{{2^{100}} - 1}
\end{array}$
Putting value of a in (3), we get $
  {2^{100}} - 1 + B = 1 \\
   \Rightarrow B = 2 - {2^{100}} \\
$
Now we know A and B so we have that value of remainder.
So our remainder is
 $
  r(x) = Ax + B \\
   = ({2^{100}} - 1)x + (2 - {2^{100}}) \\
$

Note: In such questions you have to assume remainder to be always one degree less than the divisor. If there are more than two variables in the expression of remainder then also more conditions will be given to find all of them.