Question

Find the remainder when f(x) = x3 − 14x2 51x − 22 is divided by x − 7.

Answer 1

The remainder is -8.

Answer 2

We can use at least two approaches in answering this question. But the easiest way is to use the $Remainder Theorem$

According to the Remainder Theorem, if
$(x - a)$
is not a factor of
$f(x)$
then evaluating the function at
$x = a$
gives us the Remainder when
$f(x)$
is divided by
$(x - a)$
That is
$f(a) = R$


For the given question, we have

$f(x) = {x}^{3} - 14 {x}^{2} + 51x - 22$
When this function is divided by
$x - 7$
The remainder is given by
$f(7) = R$
$f(7) = {7}^{3} - 14( {7})^{2} + 51(7) - 22$
$f(7) = 343 - 14( 49) + 51(7) - 22$
$f(7) = 343 - 686+ 357 - 22$
$f(7) = - 8$
$Hence the remainder is -8$