Introduction to Polynomials
Define polynomials and classify by number of terms (monomial, binomial, trinomial) and degree. Write polynomials in standard form, identify the leading coefficient, and evaluate polynomials for given values.
A lesson of Algebra 1, in Polynomials, about 30 minutes.
What a Polynomial Is
By the end of this lesson, you will be able to say whether an expression is a polynomial, find its degree, name it by degree and by number of terms, and write it in standard form.
A Sum of Terms with Whole-Number Powers
You have been writing polynomials since the first unit without the word.
$$3x + 5 \qquad x^2 - 7x + 2 \qquad 4a^3 - a$$
A polynomial is a sum of terms, where each term is a number times a variable raised to a whole-number power. That last condition is the whole definition, and it is what rules things out.
$$\frac{4}{x} = 4x^{-1} \qquad \text{negative power, not a polynomial}$$
$$\sqrt{x} = x^{1/2} \qquad \text{fractional power, not a polynomial}$$
$$2^x \qquad \text{the variable is the power, not a polynomial}$$
Why the restriction earns a name. Whole-number powers are exactly the expressions you can add, subtract and multiply without ever leaving the family. Divide and you can fall out of it; stick to those three operations and a polynomial stays a polynomial, which is why the next three lessons are addition, subtraction and multiplication and there is no lesson called division.
Sort It: Polynomial or Not
The last two are the pair worth remembering. $\dfrac{x}{4}$ is a polynomial and $\dfrac{4}{x}$ is not, and the difference is which one the variable is.
Degree Is the Highest Power
$$4x^3 - 2x^2 + x - 9 \quad \text{has degree } 3$$
Find the highest power on any variable term. That number is the degree, and it is the single most useful fact about a polynomial: it decides the shape of the graph, how many solutions the equation can have, and which techniques apply.
| Degree | Name | Example | Graph |
|---|---|---|---|
| $0$ | constant | $7$ | horizontal line |
| $1$ | linear | $2x + 1$ | straight line |
| $2$ | quadratic | $x^2 - 4$ | parabola |
| $3$ | cubic | $x^3 + x$ | one bend and then another |
| $4$ | quartic | $x^4 - 1$ | up to three turns |
A constant has degree zero because $7 = 7x^0$. And the degree of a term with several variables is the SUM of its powers, so $3x^2y$ has degree $3$.
Naming by Terms As Well
Two names, and a polynomial usually gets both.
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