Algebra & Polynomials

For a monic polynomial with roots $r_1,\dots,r_n$: $\sum r_i = -a_{n-1}$, $\sum_{i<j} r_ir_j = a_{n-2}$, and $\prod r_i = (-1)^n a_0$.

In plain terms. The coefficients ARE the symmetric combinations of the roots (with signs). So you can get sums and products of roots straight from the polynomial.

Example. For $x^3 - 6x^2 + 11x - 6$: the roots sum to $6$, sum in pairs to $11$, and multiply to $6$ — they are $1,2,3$.

The remainder of $P(x)$ divided by $x-a$ is $P(a)$; so $x-a$ is a factor exactly when $P(a) = 0$.

In plain terms. To find the remainder or test a factor, just plug the number in — no long division needed.

Example. $P(x) = x^2 - 5x + 6$ has $P(2) = 0$, so $x-2$ is a factor; indeed $P(x) = (x-2)(x-3)$.