If the polynomial $x^4 - 6x^3 + 16x^2 - 25x + 10$ is divided by another polynomial $x^2 - 2x + k$, the remainder comes out to be $x + a$, find $k$ and $a$.

Given:

The polynomial $x^4 - 6x^3 + 16x^2 - 25x + 10$ is divided by another polynomial $x^2 - 2x + k$, the remainder comes out to be $x + a$.

To do:

We have to find $k$ and $a$.

Solution:

Let $p(x) = x^4 – 6x^3 + 16x^2 – 25x + 10$

Remainder $= x + a$....… (i)

Dividing the given polynomial $6x^3 + 16x^2 – 25x + 10$ by $x^2 – 2x + k$, we get,

$x^2-2x+k$)$x^4-6x^3+16x^2-25x+10$($x^2-4x+8-k$

                      $x^4-2x^3+kx^2$

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

                     $-4x^3+(16-k)x^2-25x+10$

                    $-4x^3+8x^2-4kx$

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

                              $(8-k)x^2-(25-4k)x+10$

                             $(8-k)x^2-(16-2k)x+8k-k^2$

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

                                    $(-9+2k)x+10-8k+k^2$

Using equation (i), we get,

$(-9 + 2k)x + 10-8 k + k^2 = x + a$

On comparing the like coefficients, we get,

$-9 + 2k = 1$

$2k = 9+1$

$2k=10$

$k = 5$...….(ii)

$10 -8k + k^2 = a$...….(iii)

Substituting the value of $k = 5$, we get,

$10 – 8(5) + (5)^2 = a$

$10 – 40 + 25 = a$

$35 – 40 =   a$

$a =-5$

Hence, $k = 5$ and $a = -5$.

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Updated on: 10-Oct-2022

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