Question

if a polynomial x ki power 4 - 8 x cube minus 18 x square - 3 x minus Q is exactly divisible by 4 x square - 4 x + 1 then find the value of P and Q​

Answer 1

Given:

$x^{4} - 8x^{3} -18x^{2} -3x -q$   -(1)

is divisible by $4x^{2} - 4x +1$    -(2)

To find:

Value of q

Answer:

Since equation 1 is divisible by equation 2.

$4x^{2} - 4x +1$) $x^{4} - 8x^{3} -18x^{2} -3x -q$ ($\frac{x^{2} }{4} -\frac{7x}{4}+ \frac{5}{16}$

                      $x^{4} -x^{3} +\frac{x^{2} }{4}$

                     ____________________

                             $-7x^{3} - \frac{73x^{2} }{4} -3x -q$

                              $-7x^{3} +7x^{2} \frac{-7x}{4}$

                             _________________

                                           $\frac{5x^{2} }{4} -\frac{5x}{4} -q$

                                           $\frac{5x^{2} }{4} -\frac{5x}{4} + \frac{5}{16}$

We know that the reminder on division is 0.

⇒ q = -5/16

Thus, the value of q = -5/16