Question

Find the remainder when ${x^3} - a{x^2} + 6x - a$ is divided by $x - a$.

Answer 1

Hint: Substitute $x = a$ in the given polynomial. Then solve it. If the value comes out to be zero, then $x - a$ is a factor of this polynomial but if it not a factor, we get a number other that zero, which will be the remainder. We are using the implications of the Factor theorem to solve this question.

Complete step-by-step answer:
The factor theorem states that if $p(a) = 0$ then $(x - a)$ is a factor of $p(x)$. But if $p(a)$ comes out to be some other value, then that is the remainder when $p(x)$ is divided by $(x - a)$.
Substituting $x = a$ in ${x^3} - a{x^2} + 6x - a$, we get,
$
  {a^3} - a{(a)^2} + 6a - a \\
   = {a^3} - {a^3} + 5a \\
   = 5a \\
 $
Therefore, $x - a$ is not a factor and the remainder in this case is $5a$.

Note: We could also have found the remainder by the long division method which is a more intuitive method. But this method is mathematically simpler than the long division method. For the long division method, we perform a division in the traditional manner. Just that, instead of numbers, we will have polynomials.