Question

Write the function in the form f(x) = (x − k)q(x) + r for the given value of k. F(x) = x3 − x2 − 18x + 19, k = 4 f(x) = Demonstrate that f(k) = r. F(4) =

Answer 1

Solution :

Given :

$f(x)=x^3-x^2-18x+19$

By using the remainder theorem, if f(x) is divided by (x-k) and the remainder is 'r', then

$f(x)=(x-k)q(x)+r$

Here, k = 4

Therefore, divide the f(x) with (x - 4) to find the q(x) and r.

           __________________

$x-4$    |    $x^3-x^2-18x+19$         |      $x^2+3x-6$

            $x^3-4x^2$

         ------------------------------------

                    $3x^2-18x$

                    $3x^2-12x$

          -------------------------------------

                          $-6x+19$

                          $-6x+24$

          -------------------------------------

                                  $-5$

Therefore,

q(x) =  $x^2+3x-6$

  r = -5

$f(x)=(x-4)(x^2+3x-6)-5$

$f(4)=(4-4)(4^2+3 \times 4-6)-5$

       $=-5$