How do you use the remainder theorem to find the remainder for each division #(x^4-6x^2+8)div(x-sqrt2)#?

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Answer 1 · 2017-12-01T17:27:38

Remainder is #0#

let #f(x)=x^4-6x^2+8#

and #g(x)=x-sqrt(2)#

Then:#f(x)=g(x)q+r#

Where #q# and #r# are the quotient and remainder respectively.
This is The Remainder Theorem

It can be seen from this, that if we can make #g(x)=0#, then we can find the remainder #r#.

#:.#

#x^4-6x^2+8=q(x-sqrt(2))+r#

Let #x=sqrt(2)#

#(sqrt(2))^4-6(sqrt(2))^2+8=q(sqrt(2)-sqrt(2))+r#

#4-12+8=q(0)+r=>r=0#

So the remainder is #0#

There is only ever 1 remainder.