Polynomial Transformation Problem

Question

Find a polynomial with roots the same to those of $\frac{1}{2}x^2-\sqrt{7}x+2$ such that it has integer coefficients.

It seems that squaring leads to nothing and I am not sure how to approach the problem.

The roots are $$x=\sqrt{7}\pm\sqrt{3}$$ Try going from there — clathratus, Mar 12 '19 at 02:04
Does the question want the roots to be exactly the same as $\sqrt{7}\pm \sqrt{3}$, or just that $\sqrt{7}\pm \sqrt{3}$ are roots of the new polynomial? — TomGrubb, Mar 12 '19 at 02:14

score 2 · Accepted Answer · answered Mar 12 '19 at 02:20

2

Move the $\sqrt7$ term to the right, then square: $$\frac12x^2+2=\sqrt7x$$ $$\frac14x^4+2x^2+4=7x^2$$ $$\frac14x^4-5x^2+4=0$$ $$x^4-20x^2+16=0$$

answered Mar 12 '19 at 02:20

Parcly Taxel

1 Answers1