Here’s an elegant little theorem I just learned about. Informally,
A polynomial with few non-zero coefficients has few real roots.
If a polynomial has k non-zero coefficients, it can have at most 2k – 1 distinct real roots.
This says, for example, that a polynomial like x100 – 12 x37 + 1 can have at most 5 distinct real roots, even though it has a total of 100 roots when you count complex roots and count with multiplicity.
I won’t repeat the proof here, but it’s not long or advanced.
The upper bound 2k – 1 in the theorem is sharp. To see this, note that
x(x2 – 1)(x2 – 4) … (x2 – (k-1)2)
has k non-zero coefficients and 2k – 1 distinct real roots.
Source: Mathematical Omnibus
Update: Someone asked about the converse. Does having few real roots imply few non-zero coefficients? No. The polynomial (x2 + 1)n has no real roots but it has n+1 non-zero coefficients.