Polynomials Cheat Sheet

Formulae

General polynomial

p(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_{0}

a₀, a₁, …, aₙ are real coefficients and aₙ ≠ 0; the degree of p(x) is n.

Sum and product of zeros (quadratic)

\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}

For ax² + bx + c with zeros α and β; a ≠ 0.

Sum and product of zeros (cubic)

\alpha+\beta+\gamma = -\frac{b}{a}, \quad \alpha\beta+\beta\gamma+\gamma\alpha = \frac{c}{a}, \quad \alpha\beta\gamma = -\frac{d}{a}

For ax³ + bx² + cx + d with zeros α, β and γ; a ≠ 0.

Quadratic polynomial from its zeros

k\left[x^{2} - (\alpha+\beta)x + \alpha\beta\right]

A quadratic whose zeros are α and β; k is any non-zero real number.

Theorems

Division algorithm for polynomials

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then there exist polynomials q(x) and r(x) such that p(x) = g(x)·q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).

p(x) = g(x)\,q(x) + r(x)

Here p(x) = dividend, g(x) = divisor, q(x) = quotient and r(x) = remainder.

Remainder theorem

If a polynomial p(x) is divided by (x − a), then the remainder is p(a).

p(x) = (x-a)\,q(x) + p(a)

In particular, if p(a) = 0 then (x − a) is a factor of p(x) (factor theorem).

Definitions

Polynomial: An expression of the form aₙxⁿ + … + a₁x + a₀ where the powers of x are whole numbers and the coefficients are real.
Degree: The highest power of the variable in a polynomial.
Linear polynomial: A polynomial of degree 1; e.g. ax + b (a ≠ 0).
Quadratic polynomial: A polynomial of degree 2; e.g. ax² + bx + c (a ≠ 0).
Cubic polynomial: A polynomial of degree 3; e.g. ax³ + bx² + cx + d (a ≠ 0).
Value of a polynomial: The number obtained by substituting a given value of x into the polynomial; p(x) at x = k is p(k).
Zero of a polynomial: A value k for which p(k) = 0. Geometrically it is the x-coordinate of a point where the graph meets the x-axis.
Coefficient: The real number multiplying a power of the variable in a polynomial.
Constant polynomial: A polynomial of degree 0, such as 5; it has no zero unless it is the zero polynomial.

Key Points

A linear polynomial has exactly one zero; a quadratic has at most two zeros; a polynomial of degree n has at most n zeros.
The zero of a polynomial is also called its root, and equals the x-intercept of its graph.
The graph of a quadratic polynomial is a parabola: it opens upwards if a > 0 and downwards if a < 0.
A quadratic polynomial can have two distinct zeros, two equal zeros, or no real zero, depending on where its graph cuts the x-axis.
If α and β are the zeros of ax² + bx + c, then α + β = −b/a and αβ = c/a; these can be inverted to build a polynomial from its zeros.
The sum of the zeros of a quadratic is the negative of the coefficient of x divided by the coefficient of x²; the product is the constant term divided by the coefficient of x².
Geometrically, the zeros of p(x) are the x-coordinates of the points where y = p(x) intersects the x-axis.