Polynomials are expressions built from variables, coefficients, and non-negative integer exponents. They show up constantly throughout algebra and beyond, so getting comfortable with their structure and operations now will pay off in every unit that follows.
This section covers how to identify key features of polynomials (degree, leading coefficient, standard form) and how to add, subtract, and multiply them, including polynomials with more than one variable.
Polynomial Basics
Degree and leading coefficient
The degree of a polynomial is the highest exponent on any variable in the expression. It tells you a lot about the polynomial's behavior, including how many roots it can have and what its graph looks like.
- has degree 4, because the highest exponent is 4.
The leading coefficient is the coefficient attached to the highest-degree term. It affects the shape and direction of the polynomial's graph.
- In , the leading coefficient is 3 (the coefficient of the term).
A quick note on terminology by degree:
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | |
| 1 | Linear | |
| 2 | Quadratic | |
| 3 | Cubic | |
| 4 | Quartic |
Polynomial Forms and Characteristics
Standard form means writing the terms in descending order of degree, from highest to lowest. For example, if you have , standard form is .
Polynomials are also classified by the number of terms they contain:
- Monomial: one term (e.g., )
- Binomial: two terms (e.g., )
- Trinomial: three terms (e.g., )
Polynomial Operations
Addition and subtraction of polynomials
Adding and subtracting polynomials comes down to one idea: combine like terms. Like terms have the same variable(s) raised to the same exponent(s).
Addition example:
Combine , then , then .
Result:
Subtraction example:
The key step is distributing the negative sign to every term in the second polynomial first:
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Distribute the negative:
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Combine like terms:
-
Result:
Forgetting to distribute the negative to all terms in the second polynomial is one of the most common mistakes on exams. Watch for it.
Multiplication of polynomials
To multiply polynomials, use the distributive property: multiply each term in the first polynomial by every term in the second, then combine like terms.
Example:
- Combine like terms:
FOIL is a shortcut that works specifically for multiplying two binomials. It stands for First, Outer, Inner, Last:
- First:
- Outer:
- Inner:
- Last:
Combine:
FOIL only works for two binomials. For anything larger (like a binomial times a trinomial), use the full distributive property.
Operations with multiple variables
The same rules apply when polynomials have more than one variable. You just need to be more careful about identifying like terms, since both the variables and their exponents must match.
Addition example:
Result:
Multiplication example:
When multiplying terms with the same base, add the exponents (this is the product rule for exponents).
Simplification of complex polynomials
When an expression involves multiple operations, break it into manageable pieces. Work through each multiplication first, then handle addition or subtraction.
Example: Simplify
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Expand the first product:
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Expand the second product:
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Subtract the second result from the first (distribute the negative):
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Combine like terms:
Notice in step 3 that subtracting becomes . Double-check your signs whenever you distribute a negative across parentheses.