Overview

Newton's sums connect coefficients of a polynomial to power sums of its roots. They let you compute $r^k+s^k+\cdots$ without solving for roots.

Key Ideas

• For a monic quadratic $x^2 - S_1 x + S_2$, the roots satisfy $P_1 = S_1$ and $P_2 = S_1 P_1 - 2S_2$.
• For monic polynomials, the recurrence uses coefficients and earlier power sums.
• Power sums are especially useful in symmetric expressions.

Core Skills

Make the Polynomial Monic

Divide by the leading coefficient so that the Newton sum recurrence applies cleanly.

Build a Recurrence

Use the relation between coefficients and power sums to compute $P_3,P_4,\ldots$ from earlier values.

Combine with Symmetric Identities

Convert $r^k+s^k$ into expressions involving $r+s$ and $rs$ when a direct recurrence is shorter.

Worked Example

Let $x^2-5x+6=0$ have roots $r,s$. Find $r^3+s^3$.

We have $S_1 = r+s = 5$ and $S_2 = rs = 6$. First compute $P_2 = r^2+s^2 = S_1^2 - 2S_2 = 25-12=13$. Then $r^3+s^3 = (r+s)(r^2 - rs + s^2) = 5(13-6)=35$.

More Examples

Example 1: Quadratic Power Sum

If $x^2-3x+1=0$ has roots $r,s$, find $r^4+s^4$.

$P_1=3$, $P_2=P_1^2-2=7$, then $P_3=3P_2-P_1=18$, $P_4=3P_3-P_2=47$.

Example 2: Cubic Recurrence

For $x^3-2x^2+x-1=0$ with roots $r,s,t$, compute $r+s+t$ and $r^2+s^2+t^2$.

$S_1=2$, $S_2=1$, so $P_1=2$ and $P_2=S_1P_1-2S_2=2\cdot 2-2=2$.

Example 3: Symmetric Shortcut

If $r+s=4$ and $rs=3$, compute $r^3+s^3$.

$r^3+s^3=(r+s)^3-3rs(r+s)=64-36=28$.

Strategy Checklist

• Make the polynomial monic first.
• Compute $P_1, P_2$ before higher powers.
• Use symmetric shortcuts when faster.

Common Pitfalls

• Forgetting to convert to monic form first.
• Mixing $S_1,S_2$ with $P_1,P_2$.
• Skipping earlier power sums needed for the recurrence.

Practice Problems