Math Core

AMC 12 polynomial problems test a handful of big facts: Vieta's formulas for any degree, the Remainder and Factor Theorems, plugging in special values like $1$ and $-1$ , and building a new polynomial that has known roots. Once these are automatic, many "impossible-looking" polynomials fall apart in two lines.

Vieta's formulas for any degree

If $P(x) = a(x - r_1)(x - r_2)\cdots(x - r_n)$ , expanding shows each coefficient is (up to sign) a sum of products of roots. For a cubic:

Vieta for a cubic

If $r, s, t$ are the roots of $ax^3 + bx^2 + cx + d$ , then

r + s + t = -\frac{b}{a}, \qquad rs + st + tr = \frac{c}{a}, \qquad rst = -\frac{d}{a}.

The signs alternate: $-, +, -, +, \dots$ for the sums of products of $1, 2, 3, 4, \dots$ roots at a time.

As with quadratics, symmetric expressions reduce to these: $r^2 + s^2 + t^2 = (r + s + t)^2 - 2(rs + st + tr)$ and $\dfrac{1}{r} + \dfrac{1}{s} + \dfrac{1}{t} = \dfrac{rs + st + tr}{rst}$ .

Plugging into the factored form

For a monic polynomial $P(x) = (x - r)(x - s)(x - t)$ , plugging in a number $k$ gives the product $(k - r)(k - s)(k - t)$ directly. This is often faster than expanding.

Worked example: A cubic's roots

Let $r, s, t$ be the roots of $P(x) = x^3 - 4x^2 + 5x - 7$ . Find $r^2 + s^2 + t^2$ and $(1 + r)(1 + s)(1 + t)$ .

Vieta: $r + s + t = 4$ , $rs + st + tr = 5$ , $rst = 7$ . So $r^2 + s^2 + t^2 = 16 - 10 = 6$ .

For the product, $P(-1) = (-1 - r)(-1 - s)(-1 - t) = -(1 + r)(1 + s)(1 + t)$ . Since $P(-1) = -1 - 4 - 5 - 7 = -17$ , the product is $17$ . (Check with Vieta: $1 + 4 + 5 + 7 = 17$ .)

Remainders and factors

Remainder and Factor Theorems

The remainder when $P(x)$ is divided by $x - a$ is $P(a)$ . In particular, $x - a$ is a factor of $P(x)$ exactly when $P(a) = 0$ .

When dividing by a quadratic, the remainder has the form $mx + b$ . Plug in the divisor's roots to find $m$ and $b$ .

Worked example: Dividing by a quadratic

Find the remainder when $x^{100} - 3x + 2$ is divided by $x^2 - 1$ .

Write $x^{100} - 3x + 2 = (x^2 - 1)Q(x) + mx + b$ . Plug in $x = 1$ : $1 - 3 + 2 = 0 = m + b$ . Plug in $x = -1$ : $1 + 3 + 2 = 6 = -m + b$ . So $b = 3$ , $m = -3$ , and the remainder is $-3x + 3$ .

Build a polynomial with known roots

If $P(1) = 1$ , $P(2) = 2$ and $P(3) = 3$ , the polynomial $P(x) - x$ has roots $1, 2, 3$ . Knowing roots means knowing the factored form.

Worked example: Values that follow a pattern

A monic cubic $P$ satisfies $P(1) = 1$ , $P(2) = 2$ and $P(3) = 3$ . Find $P(4)$ .

$P(x) - x$ is a monic cubic with roots $1, 2, 3$ , so $P(x) - x = (x - 1)(x - 2)(x - 3)$ . Then $P(4) = 4 + 3 \cdot 2 \cdot 1 = 10$ .

Coefficient sums

The sum of the coefficients of $P$ is $P(1)$ . The alternating sum is $P(-1)$ . So

\text{sum of even-degree coefficients} = \frac{P(1) + P(-1)}{2}, \qquad \text{sum of odd-degree coefficients} = \frac{P(1) - P(-1)}{2}.

For example, the even-degree coefficients of $(x^2 + x + 1)^5$ add to $\dfrac{3^5 + 1^5}{2} = 122$ .

Common mistake

Mind the sign of the constant term in Vieta. For a cubic, the product of roots is $-\dfrac{d}{a}$ , not $\dfrac{d}{a}$ . (For even degree it's $+\dfrac{\text{constant}}{a}$ .) And Vieta uses the polynomial set equal to zero with all terms on one side.

Tip

For power sums of roots, use the equation itself. If $r$ is a root of $x^3 = x + 1$ , then $r^{n+3} = r^{n+1} + r^n$ . Summing over all roots gives a recursion for $p_n = r^n + s^n + t^n$ (Newton's sums).

Practice

Practice 1

What is the remainder when $x^5 - 2x^3 + x + 4$ is divided by $x + 2$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Let $r, s, t$ be the roots of $x^3 - 5x^2 + 3x + 2$ . What is $r^2 + s^2 + t^2$ ?

Practice 3

The polynomial $x^3 + ax^2 + bx + 6$ is divisible by both $x - 1$ and $x - 2$ . What is its third root?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

What is the sum of the coefficients of $(x^2 - 3x + 1)^7$ when it is expanded?

Practice 5

Let $r, s, t$ be the roots of $P(x) = x^3 - 3x^2 + 4x - 5$ . What is $(r + s)(s + t)(t + r)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

A monic polynomial $P$ of degree $4$ satisfies $P(1) = 2$ , $P(2) = 4$ , $P(3) = 6$ and $P(4) = 8$ . What is $P(5)$ ?

Practice 7

What is the remainder when $x^{2026}$ is divided by $x^2 + x + 1$ ?

Practice 8

Let $r, s, t$ be the roots of $x^3 - x - 1 = 0$ . What is $r^5 + s^5 + t^5$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Real contest practice

2021 AMC 12A, Problem 12: a degree-6 polynomial with positive integer roots; use Vieta to pin them down.
2021 AMC 12B Problems: a full contest for timed practice; polynomial and root questions appear throughout the AMC 12.