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If the polynomial x^4- 6x^3 + 16x^2 -26x + 10-a, is divided b

Question

If the polynomial $x^{4} - 6 x^{3} + 16 x^{2} - 26 x + 10 - a$ , is divided by another polynomial $x^{2} - 2 x + k$ the remainder comes out to be $(x + a)$ , find $k$ and $a$ .

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Henry Discussed

If the polynomial

x^{4} - 6 x^{3} + 16 x^{2} - 26 x + 10 - a

, is divided by another polynomial

x^{2} - 2 x + k

the remainder comes out to be

(x + a)

, find

k

and

a

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Text solutionVerified

\mbox {Dividend = Divisor x Quotient + Remainder}
\therefore \mbox{Dividend - Remainder = Divisor x Quotient}

x^{4} - 6 x^{3} + 16 x^{2} - 26 x + 10 - x - a

is divisible by

x^{2} - 2 x + k

Let us divide

x^{4} - 6 x^{3} + 16 x^{2} - 26 x + 10 - a

x^{2} - 2 x + k

Remainder

= 0

,
Therefore,

(- 10 + 2 k) x + (10 - a - 8 k + k^{2}) = 0

- 10 + 2 k = 0, 10 - a - 8 k + k^{2} = 0

For

- 10 + 2 k = 0

\Rightarrow 2 k = 10

\Rightarrow k = 5

For

10 - a - 8 k + k^{2} = 0

10 - a - 8 \times 5 + 25 = 0

- 5 - a = 0

a = - 5

So,

k = 5, a = - 5

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Question Text	If the polynomial $x^{4} - 6 x^{3} + 16 x^{2} - 26 x + 10 - a$ , is divided by another polynomial $x^{2} - 2 x + k$ the remainder comes out to be $(x + a)$ , find $k$ and $a$ .
Answer Type	Text solution:1
Upvotes	150