Polynomials: terms, degree and standard form

From one term to many

When two monomials aren't like terms, such as \(5x^2\) and \(-3x\), their sum can't be squeezed into a single monomial. The sum \(5x^2 - 3x\) is a new kind of object: a polynomial.

In \(5x^2 - 3x + 8\) the terms are \(5x^2\), \(-3x\) and \(8\).

Polynomials with few terms have their own names:

From four terms on, we just say polynomial.

Simplified form

A polynomial such as \(4x^3 - x + 2x^3 + 7\) contains two like terms, \(4x^3\) and \(2x^3\). Combining them gives \(6x^3 - x + 7\): same value, simpler form. Always simplify a polynomial before describing it; otherwise its degree or number of terms can come out wrong.

Degree and leading coefficient

The term with no variables is the constant term. Polynomials in one variable get special names by degree: degree 1 is linear (\(3x - 2\)), degree 2 is quadratic (\(x^2 - 5x + 6\)), degree 3 is cubic.

Standard form

\(7 - 2x^3 + x\) in standard form is \(-2x^3 + x + 7\). The leading term is \(-2x^3\) and the leading coefficient is \(-2\).

Complete and homogeneous polynomials

• \(x^3 - 4x^2 + x - 1\) is complete: \(x^3\), \(x^2\), \(x\) and the constant are all there.
• \(x^3 + 2x - 5\) isn't: the \(x^2\) term is missing.

Any missing power can be filled in with a zero coefficient: \(x^3 + 0x^2 + 2x - 5\). That placeholder becomes essential in polynomial division.

\(x^2 - 4xy + 3y^2\) is homogeneous of degree 2: every term has degree 2.

Evaluating a polynomial

A polynomial in \(x\) is often named with function notation, like \(P(x)\). Then \(P(2)\) means "the value of \(P\) when \(x = 2\)".

Practice