# If the polynomial x^4 - 6x^3 + 16x^2 - 25x + 10 is divided by x^2 - 2x + k, the remainder is x + a, find K and a.

Here ^ means raised to the power

1
by Deleted account 16.04.2015

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Here ^ means raised to the power

1
by Deleted account 16.04.2015

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P(x)
= x⁴ - 6 x³ + 16 x² - 25 x + 10

= (x²)² - x² * 6 x + 16 * x² - 25 x + 10

Instead of long division, i am doing as follows:

divisor: x² - 2 x + k

so substitute x² - 2x + k = 0 or x² = 2 x - k in P(x)

remainder = (2 x - k)² - 6 x ( 2x - k ) + 16 (2x - k) - 25 x + 10

= 4x² - 4 k x + k² - 12 x² + 6 x k + 32 x - 16 k - 25 x + 10

= - 8 x² + 2 k x + 7 x + k² - 16 k + 10

= - 8 (2 x - k) + 2 k x + 7 x + k² - 16 k + 10

=* (- 9 + 2 k ) * x +
k² - 8 k + 10*

as the remainder is x + a . compare with the above expression:

2 k - 9 = 1 and so* k = 5*

10 - 8 k + k² = a =>** a = - 5****
**

=============================

__long division__

x² - 2x + k ) x⁴ - 6 x³ + 16 x² - 25 x + 10 ( x² - 4 x + (8-k)

x⁴ - 2 x³ + k x²

======================

- 4 x³ + (16-k) x² - 25 x

- 4 x³ + 8 x² - 4 k x

================================

(8-k) x² + (4 k - 25) x + 10

8-k) x² - 2 (8-k) x + k (8-k)

===============================

(2 k - 9) x + 10 - 8 k + k²

we are given that reminder is x + a

=> 2 k - 9 = 1 hence,* k = 5*

=>**a = **10 - 8 k + k² **= - 5 ****
**

= (x²)² - x² * 6 x + 16 * x² - 25 x + 10

Instead of long division, i am doing as follows:

divisor: x² - 2 x + k

so substitute x² - 2x + k = 0 or x² = 2 x - k in P(x)

remainder = (2 x - k)² - 6 x ( 2x - k ) + 16 (2x - k) - 25 x + 10

= 4x² - 4 k x + k² - 12 x² + 6 x k + 32 x - 16 k - 25 x + 10

= - 8 x² + 2 k x + 7 x + k² - 16 k + 10

= - 8 (2 x - k) + 2 k x + 7 x + k² - 16 k + 10

=

as the remainder is x + a . compare with the above expression:

2 k - 9 = 1 and so

10 - 8 k + k² = a =>

=============================

x² - 2x + k ) x⁴ - 6 x³ + 16 x² - 25 x + 10 ( x² - 4 x + (8-k)

x⁴ - 2 x³ + k x²

======================

- 4 x³ + (16-k) x² - 25 x

- 4 x³ + 8 x² - 4 k x

================================

(8-k) x² + (4 k - 25) x + 10

8-k) x² - 2 (8-k) x + k (8-k)

===============================

(2 k - 9) x + 10 - 8 k + k²

we are given that reminder is x + a

=> 2 k - 9 = 1 hence,

=>