Package of problems

# Irrational roots Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Taking it further

1. Can you find a polynomial (not necessarily a quadratic) with integer coefficients which has $\sqrt{2}+\sqrt{3}$ as a root? What other roots (if any) does it have?

2. Can you find a polynomial with integer coefficients which has $1+\sqrt[3]{2}=1+2^{\frac{1}{3}}$ as a root? What other roots (if any) does it have?

3. Can you find a polynomial with integer coefficients which has $\sqrt[3]{2}+\sqrt[3]{4}=2^{\frac{1}{3}}+2^{\frac{2}{3}}$ as a root? What other roots (if any) does it have?

4. Can you make up some similar questions to the above?

Are there any numbers for which you cannot find a polynomial with integer coefficients which has that number as a root?