Question

Сократить дробь (x^4-2x^3+4x^2-8x+16)/(x^5+32)

Answer 1

I hope this helps you

Answer 2

To simplify the given fraction $\frac{x^4 - 2x^3 + 4x^2 - 8x + 16}{x^5 + 32}$, we can factor the numerator and denominator and then cancel out common factors.

Factor the numerator: $x^4 - 2x^3 + 4x^2 - 8x + 16$ can be factored as $x^4 - 2x^3 + 4x^2 - 8x + 16 = (x^2 - 4x + 4)(x^2 + 2x + 4)$.

Factor the denominator: $x^5 + 32$ can be factored using the sum of cubes formula, $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. In this case, $a = x$ and $b = 2$: $x^5 + 32 = (x + 2)(x^4 - 2x^3 + 4x^2 - 8x + 16)$.

Now, we can simplify the fraction by canceling the common factors: $\frac{x^4 - 2x^3 + 4x^2 - 8x + 16}{x^5 + 32} = \frac{(x^2 - 4x + 4)(x^2 + 2x + 4)}{(x + 2)(x^4 - 2x^3 + 4x^2 - 8x + 16)}$.

Notice that the factor $x^4 - 2x^3 + 4x^2 - 8x + 16$ appears in both the numerator and denominator. We can cancel it out:

$\frac{x^4 - 2x^3 + 4x^2 - 8x + 16}{x^5 + 32} = \frac{(x^2 - 4x + 4)(x^2 + 2x + 4)}{(x + 2)(x^4 - 2x^3 + 4x^2 - 8x + 16)} = \frac{x^2 - 4x + 4}{x + 2}$.

So, the simplified form of the fraction is $\frac{x^2 - 4x + 4}{x + 2}$.