Polynomials
A polynomial is an expression made of one or more terms added or subtracted. Each term is a number times variables with whole-number exponents, like 4x^3 - 2x + 7. Polynomials are named by their number of terms and their degree, which is the highest exponent.
Before this lesson: You should know algebraic expressions, combining like terms, and exponent rules.
What Is a Polynomial?
Think of polynomials as things built from small blocks. Each block is a monomial: a number, a variable, or a number times variables with whole-number exponents. Examples are 7, x, -3x^2 and 4ab^3.
Add or subtract monomials and you get a polynomial. Each monomial is called a term. The number in front of a term is its coefficient, and a term with no variable is the constant term.
Polynomial Rules: Degree, Names and Standard Form
What Counts as a Polynomial
| Polynomial | Not a polynomial | Why not |
|---|---|---|
| 3x^2 - 5x + 1 | \frac{4}{x} + 2 | A variable is in the denominator. |
| \frac{1}{2}y^3 - 7 | x^{-2} + x | A negative exponent is the same as dividing by x^2. |
| 0.4m + 6 | \sqrt{x} + 1 | A square root of a variable is not a whole-number exponent. |
| 8 | 2^x | The variable is in the exponent. That is an exponential function. |
Coefficients can be fractions, decimals or negative numbers. Only the exponents on the variables have to be whole numbers (0, 1, 2, 3, …).
Degree
The degree of a term is the exponent on its variable. If a term has more than one variable, add their exponents: 3x^2y^4 has degree 2 + 4 = 6. A constant like 8 has degree 0.
The degree of a polynomial is the highest degree of any of its terms.
Naming Polynomials
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | -6 |
| 1 | Linear | 2x + 5 |
| 2 | Quadratic | x^2 - 4x + 1 |
| 3 | Cubic | x^3 + 8 |
| 4 | Quartic | 3x^4 - x |
| 5 | Quintic | x^5 + 2x^2 |
| Number of terms | Name | Example |
|---|---|---|
| 1 | Monomial | 9x^2 |
| 2 | Binomial | 4x - 3 |
| 3 | Trinomial | x^2 + 6x + 9 |
| 4 or more | Polynomial with that many terms | x^3 - x^2 + x - 1 |
Put the two names together. For example, x^2 + 6x + 9 is a quadratic trinomial.
Standard Form
A polynomial is in standard form when its terms go from the highest degree to the lowest. The first term is the leading term, and its coefficient is the leading coefficient.
7 + 2x^3 - x \quad \longrightarrow \quad 2x^3 - x + 7
- Combine any like terms.
- Find the degree of each term.
- Write the terms from highest degree to lowest. Keep each sign with its term.
- Read off the degree, the leading coefficient and the number of terms.
Worked Examples
Example 1: Is it a polynomial?
Decide whether each expression is a polynomial.
| Expression | Polynomial? | Reason |
|---|---|---|
| 3x^2 - 5x + 1 | Yes | Every exponent is a whole number. |
| \frac{4}{x} + 2 | No | Dividing by x is the same as 4x^{-1}. |
| \frac{1}{2}y^3 - 7 | Yes | A fraction coefficient is fine. |
| x^{-2} + x | No | It has a negative exponent. |
| \frac{x}{5} + 3 | Yes | \frac{x}{5} = \frac{1}{5}x. The 5 is a number, not a variable. |
Example 2: Write in standard form and name it
Write 7 + 2x^3 - x in standard form. Then give its degree, leading coefficient and name.
The degrees are 0, 3 and 1. Order them from highest to lowest: 2x^3 - x + 7.
The degree is 3 and the leading coefficient is 2. It has 3 terms, so it is a cubic trinomial.
Example 3: Combine like terms first
Simplify 4x^2 + 3x - x^2 + 5 - 3x and name the result.
\begin{aligned} &4x^2 - x^2 + 3x - 3x + 5 \\ &= 3x^2 + 0x + 5 \\ &= 3x^2 + 5 \end{aligned}The x-terms cancel. The result has degree 2 and 2 terms, so it is a quadratic binomial. Always simplify before you name a polynomial.
Example 4: Degree with two variables
Find the degree of 3x^2y^3 - 4xy + y^2.
| Term | 3x^2y^3 | -4xy | y^2 |
|---|---|---|---|
| Degree | 2 + 3 = 5 | 1 + 1 = 2 | 2 |
The highest term degree is 5, so the polynomial has degree 5. It has 3 terms, so it is a trinomial of degree 5.
Example 5: Word problem
A ball is thrown up from a 5-foot platform. Its height in feet after t seconds is h(t) = -16t^2 + 48t + 5. Name the polynomial, and find the height after 1, 2 and 3 seconds.
It has degree 2 and 3 terms, so it is a quadratic trinomial. The leading coefficient is −16.
\begin{aligned} h(1) &= -16(1)^2 + 48(1) + 5 = -16 + 48 + 5 = 37 \\ h(2) &= -16(2)^2 + 48(2) + 5 = -64 + 96 + 5 = 37 \\ h(3) &= -16(3)^2 + 48(3) + 5 = -144 + 144 + 5 = 5 \end{aligned}The ball is 37 feet high at 1 second and again at 2 seconds, once going up and once coming down. At 3 seconds it is back to 5 feet, the height of the platform.
Common Mistakes
| Mistake | Example of the error | Fix |
|---|---|---|
| Using the number of terms as the degree | x^5 + 2 has degree 2 | The degree is the highest exponent, 5. It has 2 terms. |
| Taking the first term as the leading term | The leading coefficient of 3x + 4x^2 is 3 | Write it in standard form first: 4x^2 + 3x. The leading coefficient is 4. |
| Dropping the sign | The coefficient of -x^3 is 1 | The sign goes with the term. The coefficient is −1. |
| Using the biggest exponent in a term with two variables | x^2y^3 has degree 3 | Add the exponents: 2 + 3 = 5. |
| Calling every expression a polynomial | \frac{6}{x} - 1 is a binomial | A variable in the denominator means it is not a polynomial. |
| Naming before simplifying | x^2 + 3x - x^2 is a quadratic trinomial | Combine like terms first: it is 3x, a linear monomial. |
Polynomials as Functions
You can write a polynomial with function notation, like p(x) = x^2 - 4x + 1. The degree tells you a lot about the graph:
- Degree 1: the graph is a straight line, as in slope-intercept form.
- Degree 2: the graph is a U-shape called a parabola. You saw one in key features of functions.
- Degree 3 or more: the graph can turn more than once.
Don’t mix up polynomials with exponential functions. In x^2 the variable is the base. In 2^x the variable is the exponent, so it is not a polynomial.
Practice Problems
- Is 6x^4 - x + 2 a polynomial? If so, name it.
- Is 3x^{-1} + 4 a polynomial?
- Is 0.5m^2 - \frac{3}{4} a polynomial? If so, name it.
- Write 9 - 4x^2 + x^5 in standard form. Give the degree and leading coefficient.
- Write 3x - 8x^3 + 2 - x^2 in standard form. Give the degree, leading coefficient and number of terms.
- Simplify 5x^3 + 2x - 5x^3 + 7 and name the result.
- Find the degree of 4a^3b^2 + 2ab^4 - 9.
- What is the degree of the constant -12?
- Evaluate p(x) = 2x^3 - x^2 + 4 at x = -2.
- A box has volume V(x) = 4x^3 - 36x^2 + 80x cubic inches. Name the polynomial and find V(2).
Show answers
- Yes. It has degree 4 and 3 terms, so it is a quartic trinomial.
- No. It has a negative exponent. 3x^{-1} means \frac{3}{x}.
- Yes. Decimal and fraction coefficients are fine. It is a quadratic binomial.
- x^5 - 4x^2 + 9. Degree 5, leading coefficient 1.
- -8x^3 - x^2 + 3x + 2. Degree 3, leading coefficient −8, 4 terms.
- 2x + 7. The x^3 terms cancel, so it is a linear binomial.
- The term degrees are 5, 5 and 0. The degree is 5.
- 0. A constant has no variable, so its degree is 0.
- 2(-8) - 4 + 4 = -16. So p(-2) = -16.
- It is a cubic trinomial. V(2) = 4(8) - 36(4) + 80(2) = 32 - 144 + 160 = 48, so the volume is 48 cubic inches.
FAQ
What is a polynomial in simple terms?
A polynomial is a sum of terms like 3x^2, -5x and 7. Each term is a number times variables with whole-number exponents.
Is a single number a polynomial?
Yes. A number like 8 is a monomial with degree 0. It is called a constant.
How do you find the degree of a polynomial?
Find the degree of each term, adding exponents if a term has more than one variable. The highest one is the degree of the polynomial.
Why can’t a polynomial have a negative exponent?
A negative exponent means dividing by the variable, like x^{-1} = \frac{1}{x}. Polynomials only multiply variables, so their exponents must be 0, 1, 2, 3, and so on.
Summary
- A polynomial is one or more terms added or subtracted, with whole-number exponents on the variables.
- The degree is the highest exponent. For a term with two variables, add the exponents.
- Name it by degree (constant, linear, quadratic, cubic) and by number of terms (monomial, binomial, trinomial).
- Standard form lists terms from highest degree to lowest. The first coefficient is the leading coefficient.
- Simplify first, then name the polynomial.
Previous lesson: Linear vs. Exponential Functions
Next lesson: Adding and Subtracting Polynomials
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