Question

X^5-4*x^4-4*x+16/(x^2-2)*(x^2-5*x+4)

Answer 1

(x^5-4x^4-4x+16)/(x^2-2)(x-1)(x-4)=(x^3-4x^2+2x-8)/(x-1)(x-4)=(x-4)(x^2+2)/(x-4)(x-1)=(x^2+2)/(x-1)

Answer 2

To simplify the expression, we can factor the numerator and denominator first:

Numerator:

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x^5 - 4x^4 - 4x + 16

= x^4(x - 4) - 4(x - 4)

= (x^4 - 4)(x - 4)

= (x^2 + 2)(x^2 - 2)(x - 4)

Denominator:

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(x^2 - 2)(x^2 - 5x + 4)

= (x - sqrt(2))(x + sqrt(2))(x - 1)(x - 4)

Now we can cancel out the common factors in the numerator and denominator:

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(x^2 + 2)(x^2 - 2)(x - 4)
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(x - sqrt(2))(x + sqrt(2))(x - 1)(x - 4)

= (x^2 + 2)(x^2 - 2)
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(x - sqrt(2))(x + sqrt(2))(x - 1)

= [(x + sqrt(2))(x - sqrt(2))(x^2 + 2)] / [(x - 1)(x + sqrt(2))]

= (x^2 - 2) / (x - 1)

Therefore, the simplified expression is (x^2 - 2) / (x - 1).