## Task B-1.4.

Simplify the following fraction to the one that cannot be further reduced

$$
\frac{x^{4}-16}{x^{4}-4 x^{3}+8 x^{2}-16 x+16}
$$
## Solution.

Let's factorize the numerator and the denominator of the given algebraic fraction.

$$
\begin{array}{rlrl}
x^{4}-16=\left(x^{2}-4\right) & \left(x^{2}+4\right)=(x-2)(x+2)\left(x^{2}+4\right) & & 2 \text { points } \\
x^{4}-4 x^{3}+8 x^{2}-16 x+16 & =\left(x^{4}-4 x^{3}+4 x^{2}\right)+\left(4 x^{2}-16 x+16\right) & \\
& =x^{2}(x-2)^{2}+4(x-2)^{2} & & 2 \text { points } \\
& =\left(x^{2}+4\right)(x-2)^{2} & & 1 \text { point }
\end{array}
$$

Therefore,

$$
\frac{x^{4}-16}{x^{4}-4 x^{3}+8 x^{2}-16 x+16}=\frac{(x-2)(x+2)\left(x^{2}+4\right)}{\left(x^{2}+4\right)(x-2)^{2}}=\frac{x+2}{x-2}
$$