How many real roots does this polynomial have?
$$x^4 – 4x^3 + 4x^2 – 10$$
Because non-real roots come in pairs, it must have 4, 2 or 0 real roots.
Following Descartes' rules of signs, it either has one negative (real) number and one or three positive numbers.
How can I tell if it has 2 or 4 real roots?
Thank you very much in advance.
Best Answer
Hint:
Your polynomial can be factored into two quadratics using difference of squares:
$$x^2(x-2)^2-10=(x(x-2)+\sqrt{10})(x(x-2)-\sqrt{10}).$$
Can you take it from here?