Prove that if then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4$ is divisible by $(x – x_0)^2$.
Let $f (x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4$. By hypothesis, $f (x_0) = f ‘(x_0) = 0$. Consequently, $x_0$ is the double root of the polynomial $f (x)$, that is, the polynomial $f (x)$ is divisible by $(x – x_0)^2$.
See the solution above.
Find the coefficient of x for the polynomial $($x – a$)$ $($x – b$)$ $($x – c$)$ … $($x – z$)$.
Among the factors there is a bracket $($x – x$)$.
Derive from the theorem in question 61013 that
is an irrational number.
The indicated number is the root of the polynomial $x^2$ – 17. According to Problem 61013, all the rational roots of this polynomial are integers. But this equation obviously does not have any roots.