polynomial
A polynomial is built like a tower of blocks, where each block is a number times a variable raised to a whole-number power, and you stack them together with plus and minus signs. Something like 3x^2 - 5x + 7 is a polynomial; it is a controlled, well-behaved expression with no surprises like dividing by the variable or taking its square root.
More precisely, a polynomial in one variable is a sum of terms of the form (coefficient) times (variable raised to a non-negative integer power). The powers must be whole numbers: 0, 1, 2, 3, and so on. So 4x^3, -x, and the lone number 6 are all polynomials, and so is their sum 4x^3 - x + 6.
The honest caveat is what a polynomial is NOT. Expressions like 1/x (which is x^-1), sqrt(x) (which is x^(1/2)), or 2^x are not polynomials, because their exponents are negative, fractional, or have the variable in the exponent. Polynomials are deliberately the simplest large family of expressions, which is exactly why so much of algebra is about understanding them.
3x^2 - 5x + 7 is a polynomial; 3/x + 5 is not (the variable sits in a denominator).
Whole-number powers only — no variable in the denominator or under a root.
The word comes from poly- (many) and -nomial (terms or names), literally a many-termed expression. A single number with no variable, like 5, counts as a polynomial too — it is just a constant polynomial.