Question

If $a{x^3} + b{x^2} + c$ is divided by $(x - 3)$ , then find the remainder.
A) (-27a + 9b+c)
B) (-27a-9b)
C) (27a + 9b + c)
D) (27a + 9b)
E) (-27a+9b)

Answer 1

- Hint: Remainder theorem: Let f(x) be any polynomial with degree greater than or equal to 1 and let $\alpha $ be any real number. If f(x) is divided by $(x - \alpha )$ , then the remainder is $f(\alpha )$.
Also, the remainder $f(\alpha )$ is equal to the value of the polynomial at $x = - \alpha $.

Complete step-by-step solution -
Let f(x) = $a{x^3} + b{x^2} + c$ and f(x) is divided by $(x - 3)$ then by the Remainder theorem we can say that f(3) is the remainder.
Now in order to calculate the remainder i.e., f (3) we will substitute x= 3 in expression $a{x^3} + b{x^2} + c$
Therefore, f (3) = $a{(3)^3} + b{(3)^2} + c$
After solving the exponent part in the above equation our equation will become f (3) = 27a + 9b + c.
Hence, the remainder of $a{x^3} + b{x^2} + c$ is 27a + 9b + c, when divided by $(x - 3)$

Hence the option C) is correct.

Additional Information:
 We can make use of remainder theorem to find the remainder of the given polynomial if the divisor is a linear polynomial.
If a polynomial $f(x)$ is divided by \[(x + \alpha )\], then remainder = $f( - \alpha )$ = the value of the polynomial at $x = \alpha $.
If a polynomial $f(x)$ is divided by ax + b, $a \ne 0$ , then the remainder = $f\left( { - \dfrac{b}{a}} \right)$ = the value of the polynomial $f(x)$ at $x = \dfrac{b}{a}$.

Note:
We can find remainder by making use of division algorithm for polynomials which states if a polynomial f(x) is divided by a non-zero polynomial g(x) then there exist unique polynomials q(x) and r(x) such that f(x)= g(x) q(x) + r(x) where either r(x) = 0 or deg r(x) $ < $ deg g(x) where dividend = f(x), divisor = g(x), quotient = q(x) and remainder = r(x) but remainder theorem is the shortest and simplest method to find the remainder of any polynomial which is divided by a linear polynomial.