If $x=\sqrt2+\sqrt3+\sqrt6$ is a root of $x^4+ax^3+bx^2+cx+d = 0$ then what is the value of $|a+b+c+d|$?
My first thought was to substitute $x$ in equation. But that way is very lengthy. I looked for any other way to approach it. But I could not find a reasonable approach. Any suggestions??
If you can use linear algebra, try this.
Write $x\cdot1$, $x\cdot \sqrt2$, $x\cdot \sqrt3$, $x\cdot \sqrt6$ as integer linear combinations of $1,\sqrt2,\sqrt3,\sqrt6$. Write these integers as columns in a matrix $A$. Find the determinant of $A-I$. This is $f(1)$, where $f$ is the characteristic polynomial of $A$, which is monic of degree $4$ and has $x$ as a root.