An equation with roots of sqrt(5) + sqrt(7)

My older son is working thorugh the Integrated CME Project Mathematics III book this summer. Last week he came across a pretty interesting problem in the first chapter of the book.

That chapter is about polynomials and the question was to find a polynomial with integer coefficients having a root of $\sqrt{5} + \sqrt{7}$. The follow up to that question was to find a polynomial with integer coefficients having a root of $3 + \sqrt{5} + \sqrt{7}$.

His original solution to the problem as actually terrific. His first thought was to guess that the solution would be a quadratic with second root $\sqrt{5} - \sqrt{7}$. That didn’t work but it gave him some new ideas and he found his way to the solution.

Following his solution, we talked about several different ways to solve the problem. Earlier this week we revisited the problem – I wanted to make sure the ideas hadn’t slipped out of his mind.

Here’s how he approached the first part:

Here’s the second part:

Finally, we went to Mathematica to check that the polynomials that he found do, indeed, have the correct numbers as roots.

I like this problem a lot. It is a great way for kids learning algebra to see polynomials in a slightly different light. They also learn that solutions with square roots are not automatically associated with quadratics!