Finding Zeroes of Polynomials

The rational root theorem is simply a way to make a short list of guesses for a polynomial's zeros. "Rational" means a fraction of whole numbers, and "root" is another word for zero, so it's a theorem about which fractions could possibly be zeros. The formula has three small pieces: $p$ (a number that divides the constant term $a_0$), $q$ (a number that divides the leading coefficient $a_n$), and $\frac pq$ (a candidate). You try the candidates, and that's the whole method.

Where we are. The last few lessons showed that every zero $r$ peels off a factor $x - r$. The catch is finding that first zero. For a quadratic there's a formula; for higher degrees you mostly have to guess. This theorem tells you where to guess.

The process:

1. List all $\pm\frac pq$.
2. Test candidates by evaluating $P$ or by synthetic division until one gives zero.
3. Divide it out: the quotient has degree one lower.
4. Repeat until a quadratic remains, then use the quadratic formula (complex zeros are possible).

Why it works. Suppose $\frac pq$, in lowest terms, is a zero. Plug it in and multiply through by $q^n$ to clear the fractions. Every term except $a_0q^n$ has a factor of $p$, so $p$ must divide $a_0q^n$, and since $p$ shares nothing with $q$, it must divide $a_0$. The same argument from the other end, with every term except $a_np^n$ having a factor of $q$, shows that $q$ divides $a_n$.

A common mistake. It's tempting to think the theorem finds all the zeros. It only lists the possible rational ones. A zero like $\sqrt2$ or $1 + i$ won't be on the list, which is why step 4 finishes with the quadratic formula.

Computer algebra systems start with exactly this search before falling back to numerical methods like Newton's method for irrational zeros.