We can use at least two approaches in answering this question. But the easiest way is to use the [tex]<b>Remainder Theorem</b>[/tex]
According to the Remainder Theorem, if
[tex](x - a)[/tex]
is not a factor of
[tex]f(x)[/tex]
then evaluating the function at
[tex]x = a[/tex]
gives us the Remainder when
[tex]f(x)[/tex]
is divided by
[tex](x - a)[/tex]
That is
[tex]f(a) = R[/tex]
For the given question, we have
[tex]f(x) = {x}^{3} - 14 {x}^{2} + 51x - 22[/tex]
When this function is divided by
[tex]x - 7[/tex]
The remainder is given by
[tex]f(7) = R[/tex]
[tex] f(7) = {7}^{3} - 14( {7})^{2} + 51(7) - 22[/tex]
[tex] f(7) = 343 - 14( 49) + 51(7) - 22[/tex]
[tex] f(7) = 343 - 686+ 357 - 22[/tex]
[tex] f(7) = - 8[/tex]
[tex]<b>Hence the remainder is -8</b>[/tex]
