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
\(f(7) = R\)
\(f(7) = {7}^{3} - 14( {7})^{2} + 51(7) - 22\)
\(f(7) = 343 - 14( 49) + 51(7) - 22\)
\(f(7) = 343 - 686+ 357 - 22\)
\(f(7) = - 8\)
\(Hence the remainder is -8\)
