Definition: A polynomial is an expression made of terms that are numbers times variables raised to whole-number powers, added or subtracted, such as 4x^3 – 7x + 2.
What counts as a polynomial?
Every term has a coefficient and a variable part with an exponent that is 0, 1, 2, 3, and so on. No variable may sit in a denominator, under a root sign, or in an exponent. So 5x2 + 3x – 8 is a polynomial, but 4 / x and the square root of x are not.
How do you read one?
Terms are separated by plus or minus signs. Standard form lists them from the highest power down. Like terms, those with the same variable part, can be combined.
Example. Simplify (6x2 – 4x + 9) – (2x2 + 5x – 3). Distribute the minus sign: 6x2 – 4x + 9 – 2x2 – 5x + 3. Combine like terms to get 4x2 – 9x + 12.
What goes wrong?
The classic slip is to subtract only the first term of the second polynomial, leaving the other signs unchanged. Another is to combine x and x2, which are different kinds of terms. Polynomials are closed under addition, subtraction, and multiplication.