Algebra › Polynomials

Polynomials

A polynomial is a sum of terms — each one a number times a power of x. Add them, multiply them, and the answer is still a polynomial.

Multiply two binomials

FOIL: First, Outside, Inside, Last.

(ax + b)(cx + d)

(x + 2)(x + 3)

First

1x · 1x = 1x²

Outside

1x · 3 = 3x

Inside

2 · 1x = 2x

Last

2 · 3 = 6

Combine like terms

= x^2 + 5x + 6

What counts as a polynomial

A polynomial in x is built from terms like aₙxⁿ, where the powers are whole numbers (0, 1, 2, …) and the coefficients are ordinary numbers. Examples:

3x + 2 — a linear polynomial (degree 1).
x² − 5x + 6 — a quadratic (degree 2).
4x³ − x + 7 — a cubic (degree 3).

The degree is the highest power. The leading coefficient is the number in front of that highest-power term.

Not a polynomial

√x, 1/x, 2ˣ — none of these are polynomials. Polynomials don't have variables under roots, in denominators, or as exponents.

Adding and subtracting

Combine like terms — same variable, same power. Example:

(3x² + 2x − 1) + (x² − 5x + 4) = 4x² − 3x + 3

Multiplying — every-by-every

Each term in the first polynomial multiplies each term in the second. For two binomials this is FOIL:

First — the leading terms.
Outside — first of the first, last of the last.
Inside — last of the first, first of the last.
Last — the trailing terms.

Special products to memorise

Square of a sum — (a + b)² = a² + 2ab + b²
Square of a difference — (a − b)² = a² − 2ab + b²
Difference of squares — (a + b)(a − b) = a² − b²

Quick check

What is the degree of 5x⁴ − x² + 7?
Expand (x + 5)(x − 5).
Add (2x² − x) + (x² + 3x − 4).

Answers: 4, x² − 25, and 3x² + 2x − 4.

Quick check

What is the degree of 5x⁴ − x² + 7?

Quick check

Expand (x + 5)(x − 5).