What reduced quadratic equation has solutions 2 and 3?
What reduced quadratic equation has solutions and ?
Answer & Explanation
A polynomial with coefficients in that has as a root must also have as a root; likewise, if it has as a root, then it must also have as a root.
So any polynomial with coefficients in that has both and as roots must be of degree at least 4 or equal to 0, and must be a multiple of .
So there is no quadratic polynomial with coefficients in that will do it.
To see this, suppose that p(x) is a polynomial with coefficients in that has as a root. Dividing with remainder by we get
where q(x) and r(x) have rational coefficients, and or else r(x) is of degree 0 or 1.
Plugging in , we get
But there are no polynomials of degree 0 or 1 with coefficients in that are 0 at (that would yield that is rational), so .
Therefore, , so p(x) is a multiple of ; and since is 0 at , then so is p(x). The same argument shows that it is also a multiple of , and so p(x) will also have a a root.
Since and are relatively ' in , any polynomial that is a multiple of both is a multiple of their product. The smallest degree polynomial that doe is
If you allow other coefficients, then the only quadratics that work are the form
and its multiples.