Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term"). Polynomial refers to "many terms" exoressions.

A polynomial can have:

• constants (like 3, -20, or ½)
• variables (like x and y)
• exponents (like the 2 in y²), but only 0 abd positive integer exponents are allowed

Polynomials can be combined using addition, subtraction, multiplication and division. However, polynomials cannot be combined by division. For example, something like 2/x is not a polynomial.

These are polynomials:

• 3x
• x - 2
• -6y² - (7/9)x
• 3xyz + 3xy²z - 0.1xz - 200y + 0.5
• 512v⁵+ 99w⁵
• 5

These are not polynomials:

•  3xy^-2 is not, because the exponent is "-2"
•  2/(x+2) is not, because dividing by a variable is not allowed
•  1/x is not because exponent of variable x is -1
•  √x is not, because the exponent of variable x is "½"

These are polynomials:

• x/2, because you can divide by a constant
• 3x/8, because you can divide by a constant
• √2, because it is a constant

Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms.

Polynomials can have no variable at all. For example, 56 is a polynomial. It has just one term, which is a constant.

Polynomials can have one variable. For example: x⁴-2x²+x has three terms, but only one variable (x).

Polynomials can have two or more variables. For example: xy⁴-5x²z has two terms, and three variables (x, y and z).

If you add polynomials you get a polynomial; if you multiply polynomials you get a polynomial. But if you divide polynomials by another polynomial, you may or may not end up with a polynomial.

The degree of a polynomial with only one variable is the largest exponent of that variable. For example: In 4x³-x-3, The Degree is 3 (the largest exponent of x).

The Standard Form for writing a polynomial is to put the terms with the highest degree first. For example: The Standard Form of polynomial: 3x² - 7 + 4x³ + x⁶ is x⁶ + 4x³ + 3x² - 7

Like Terms are terms whose variables (and their exponents such as the 2 in x²) are the same. In other words, terms that are "like" each other. (Note: the coefficients can be different). For example:

(1/3)xy²     -2xy²     6xy²

Are all like terms.

To simplify a polynomial, you usually combine all the like terms.

Polinomial in different formats

If f(x/3) = 3x^2 + x + 1
then
f(x) = 9x^2+3x+1
f(3x) = 81x^2+9x+1
f(3x) - 7 = 81x^2+9x+1-7 = 81x^2+9x-6