<s>[INST] How to solve $x^4-8x^3+24x^2-32x+16=0$ How can we solve this equation?
$x^4-8x^3+24x^2-32x+16=0.$ 
 [/INST] To start, this polynomial is called a biquadratic equation.
First, factor out the polynomial.
$$x^4 - 2x^3 - 6x^3 + 12x^2 + 12x^2 - 24x - 8x + 16 = 0$$
$$x^3(x - 2) - 6x^2(x - 2) + 12x(x - 2) - 8(x - 2) = 0$$ 
$$(x^3 - 6x^2 + 12x - 8)(x - 2) = 0$$
Then after, continue the factorization of the polynomial.
$$(a + b)^2 = a^2 + 2ab + b^2$$ 
$$(a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$$ 
$$(a + b)^4 = a^4 + 4a^3b + 6a^2 b^2 + 4ab^3 + b^4$$ 
$$x^4 + 4 \times x^2 \times (-2) + 6 \times x^2 \times (-2)^2 + 4 \times x \times (-2)^3 + (-2)^4 = 0$$ 
So $\mathbf{(x - 2)^4 = 0}$. 
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