show that ifxar br2 cr d then f kr where r ak bk2 ck d

show that if/(x)-ar, br2 + cr + d then f (k)-r, where r = ak, + bk2 + ck + d , using long division. In other words, verify the Remainder Theorem for a third-degree polynomial function.

Solution

f(x) = ax^3 + bx^2 + cx + d

by remainder theorem dividing the polynomial by ( x- k )

( ax^3 + bx^2 + cx + d ) / ( x- k)

on dividing by long division method

we get

quotient as ( ax^2 + x (b+ak) + k (b + ak ) + c) )

remainder = ak^3 + bk^2 + ck + d

remainder theorem proved