maths polynomials

General form of a polynomial in one variable

p(x) = a extsubscript{n}x extsuperscript{n} + a extsubscript{n-1}x extsuperscript{n-1} + … + a extsubscript{1}x + a extsubscript{0}, where a extsubscript{n}

\neq 0 and n is a non-negative integer.

Degree of a polynomial

The highest power of the variable in the polynomial.

Degree of a non-zero constant polynomial

0

Degree of the zero polynomial

Not defined

Monomial

A polynomial with one term.

Binomial

A polynomial with two terms.

Trinomial

A polynomial with three terms.

Linear polynomial

A polynomial of degree 1. General form

Quadratic polynomial

A polynomial of degree 2. General form

Cubic polynomial

A polynomial of degree 3. General form

Zero of a polynomial

A real number c such that p(c) = 0.

Zero of a linear polynomial p(x) = ax + b

x = -b/a, where a

\neq 0.

Number of zeroes of a linear polynomial

One and only one zero.

Non-zero constant polynomial has

No zero.

Zeroes of the zero polynomial

Every real number is a zero.

Factor Theorem

(x – a) is a factor of p(x) if and only if p(a) = 0.

Identity I

x

\textsuperscript{2} + 2xy + y

\textsuperscript{2}

Identity II

x

\textsuperscript{2} – 2xy + y

\textsuperscript{2}

Identity III

(x + y)(x – y)

Identity IV

x

\textsuperscript{2} + (a + b)x + ab

Identity V

x

\textsuperscript{2} + y

\textsuperscript{2} + z

\textsuperscript{2} + 2xy + 2yz + 2zx

Identity VI

x

\textsuperscript{3} + y

\textsuperscript{3} + 3xy(x + y) or x

\textsuperscript{3} + 3x

\textsuperscript{2}y + 3xy

\textsuperscript{2} + y

\textsuperscript{3}

Identity VII

x

\textsuperscript{3} – y

\textsuperscript{3} – 3xy(x – y) or x

\textsuperscript{3} – 3x

\textsuperscript{2}y + 3xy

\textsuperscript{2} – y

\textsuperscript{3}

Identity VIII

(x + y + z)(x

\textsuperscript{2} + y

\textsuperscript{2} + z

\textsuperscript{2} – xy – yz – zx)

If x + y + z = 0, then x

\textsuperscript{3} + y

\textsuperscript{3} + z

\textsuperscript{3} = ?

3xyz

Coefficient of a term

The numerical factor of the term (e.g., in –x

\textsuperscript{3} , the coefficient is –1).

Constant polynomial

A polynomial with only a constant term (e.g., 2, –5, 7).

Zero polynomial

The constant polynomial 0.

Value of a polynomial p(x) at x = a

Found by substituting a for x in the polynomial.

How to check if (x + 1) is a factor of p(x)

Find p(–1). If p(–1) = 0, then (x + 1) is a factor.

Method to factorise ax

\textsuperscript{2} + bx + c

Split the middle term bx into two terms whose product is ac.

For cubic polynomials, first step in factorisation

Find a zero a such that p(a) = 0, then (x – a) is a factor.