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without using actual division method, find the quotient and remainder when $x^6-2x^4+x^2+5$ is divided by $x^2-2$.

Using Remainder Theorem I got the remainder as $7$ but how do I get the quotient..

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    Processing two polynomials in order to get their quotient is called a division operation. So it will be very very hard to compute the quotient without... dividing. – 2017-01-27

2 Answers 2

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Let $\,X = x^2$ then $\,f-7 = X^3-2X^2+X-2 = (\color{#c00}{X-2})X^2 + \color{#c00}{X-2}$ is easy to divide by $\,\color{#c00}{X-2}$

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$x^6-2x^4+x^2+5=x^4(x^2-2)+x^2-2+7=(x^4+1)(x^2-2)+7$

so quotient is $x^4+1$