Tutorial: Polynomial Vocabulary

To play with polynomials it’s essential to have the right vocabulary. Today we’ll define term, coefficient, variable, degree, polynomial, standard form, leading coefficient, constant termmonomial, binomial, and trinomial.


Term:

Terms are the building blocks of polynomials. Each term is made up of a coefficient, a variable, and a degree. Some example terms: 3x^2, x^5, -2x^{12}, \frac{1}{2}x, 17. Notice that terms do not include plus or minus signs, only multiplication. A solo term is a special kind of polynomial called a monomial.

Coefficient:

The coefficient of a term is the number. So, for 3x^2, 3 is the coefficient. For x^5, the coefficient is invisible – you’re multiplying x^5 by 1 – 1 is the coefficient. If your term is 17, the coefficient is 17.

Variable:

Variables, as you probably already know, are letters that represent numbers. The letter we use as a variable most often is x.

Degree:

The degree of a term is equal to the number that the variable is raised to. If the term is 3x^2, the degree is 2. For the term 17, the variable x is missing. The only power you can raise x to where it will disappear is x^0=1, so the degree is 0. Notice that the degree is never negative, and never a fraction – these types of degrees are not allowed in polynomials.

When you are asked to combine like terms, you add or subtract the coefficients of terms that have the same degree. For example, 3x^4-8x^4 = -5x^4.


Polynomial:

A polynomial is a collection of terms connected by addition. You can create a polynomial by combining all of the terms introduced above: 3x^2+x^5+-2x^{12}+\frac{1}{2}x+17. To put a polynomial in standard form, first combine like terms (none here) and then sort terms by degree – highest first: 2x^{12}+x^5+3x^2+-\frac{1}{2}x+17. You will usually see the addition of a negative term simplified to subtraction: 2x^{12}+x^5+3x^2-\frac{1}{2}x+17.

Degree of a Polynomial:

The degree of a polynomial is equal to the degree of its highest degree term. This means that 2x^{12}+x^5+3x^2-\frac{1}{2}x+17 has a degree of 12.

Leading Coefficient:

The leading coefficient of a polynomial is the coefficient of its highest degree term. So 2x^{12}+x^5+3x^2-\frac{1}{2}x+17 has a leading coefficient of 2.

Constant Term:

The constant term of a polynomial is the term (if there is one) with no x‘s, also known as the term with degree 0. For 2x^{12}+x^5+3x^2-\frac{1}{2}x+17, the constant term is 17.


Special Names:

Monomial: A polynomial with only one term.

Binomial: A polynomial with two terms. “First degree binomial” is a binomial with an x term and a constant term: 3x-4 or x+10.

Trinomial: A polynomial with three terms. You’ll hear about “second degree trinomials,” which are quadratic expressions with three terms: 2x^2-3x+10.

Polynomials with more than three terms are just called polynomials; there aren’t any (commonly used) special names.


Other posts for playing with polynomials:

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