Polynomials: Key Concepts, Examples, and Study Guide
In algebra, a polynomial is a sum or difference of terms like a·xⁿ, where every exponent n is a whole number. Read the degree as the highest power of the variable, list the terms in descending order, and add or subtract by combining like terms. Multiply term by term, pull out the greatest common factor before factoring, then join the factors to the zeros and multiply back to check the signs.
What Is a Polynomial?
A polynomial is an expression built by adding or subtracting terms. Each term is a number times a variable raised to a non-negative integer power, such as 3x² or -5x. The number in front of a variable is a coefficient, and a term with no variable, like 7, is a constant.
An expression stops being a polynomial when a variable breaks that pattern:
- A negative exponent, such as x⁻¹, which is the same as 1/x.
- A fractional exponent, such as x^(1/2), which works like a root.
- A variable sitting in a denominator, such as 1/(x+1).
- A variable sitting under a root sign, such as √x.
The visual below splits the two sides of this test.
OpenStax writes a polynomial as aₙxⁿ+⋯+a₂x²+a₁x+a₀, where each aᵢ is a coefficient. That single condition — whole-number exponents on the variable — is what separates polynomials from rational expressions and radical expressions.
Degree, Leading Term, and Standard Form
The degree of a polynomial is the highest power of the variable that appears. The term with that power is the leading term, and its coefficient is the leading coefficient. Listing the terms from highest power to lowest puts the polynomial in standard form.
Two edge cases trip people up:
- x alone has degree 1, because x = x¹.
- A nonzero constant, such as 6, has degree 0, because it is 6·x⁰.
Take 4x⁵ - 7x² + x. The highest power is 5, so the degree is 5, the leading term is 4x⁵, and the leading coefficient is 4. In standard form, 4x⁵ stays first, followed by the lower powers. The leading term also sets the long-run shape of the graph, which is why it gets its own name.
Monomials, Binomials, and Trinomials
These names count terms, not degree:
Name | Number of terms | Example |
Monomial | 1 | 5x⁴ |
Binomial | 2 | 2x - 9 |
Trinomial | 3 | -3x² + 8x - 7 |
A binomial might be degree 1 or degree 10; the name only reports how many terms are present. Reach for the degree when you need the highest power, and reach for monomial, binomial, or trinomial when you need to choose a factoring or multiplication pattern.
Adding, Subtracting, and Multiplying
Add and subtract by combining like terms: terms that share the same variable raised to the same power. Unlike terms stay put.
- Add: (3x² + 2x - 1) + (5x² - 4x) = 8x² - 2x - 1.
- Subtract: (3x² + 2x - 1) - (5x² - 4x) = -2x² + 6x - 1. Distribute the minus sign to every term in the second parentheses.
Multiply with the distributive property: multiply every term in the first polynomial by every term in the second, then combine like terms.
- 2x(3x² - x + 4) = 6x³ - 2x² + 8x.
- For two binomials, FOIL keeps the four products straight: (x + 3)(x - 2) = x² - 2x + 3x - 6 = x² + x - 6.
Two patterns recur often enough to memorize:
Pattern | Expansion |
Square of a sum or difference | (a ± b)² = a² ± 2ab + b² |
Difference of squares | (a + b)(a - b) = a² - b² |
With monomials like (2x³)(5x²), add the exponents on the shared base and multiply the coefficients. Brush up on the exponent rules before combining degree and coefficient in polynomial multiplication. The degree of a product is the sum of the factor degrees, so a degree-2 times a degree-3 polynomial gives a degree-5 product.
Factoring: Choose the Right Method
Factoring writes a polynomial as a product of simpler polynomials. This section gives the method overview; the step-by-step trinomial working lives in the linked guide below.
- Pull out the greatest common factor first. Always, even when a special pattern looks ready-made.
- Match the structure to a method.
Structure | Method | Result |
Two terms, difference of squares | a² - b² | (a + b)(a - b) |
Three terms, ax² + bx + c | Trinomial factoring | Two binomial factors |
Four terms | Grouping | Two grouped binomial factors |
Perfect-square trinomial | a² ± 2ab + b² | (a ± b)² |
With x² - 9, spot the difference of squares: x² - 9 = (x + 3)(x - 3). For a quadratic trinomial like x² + 5x + 6, the full step-by-step working is in the how to factor a quadratic guide. After any factoring, multiply the factors back and compare signs, because sign slips are where points most often disappear.
Polynomial Functions: Zeros, Factors, and Graphs
Write a polynomial as P(x) and it becomes a function. A value c with P(c) = 0 is a zero, and the Factor Theorem says (x - c) is a factor exactly when P(c) = 0. For P(x) = x² - 3x + 2 = (x - 1)(x - 2), the zeros are 1 and 2, and the graph meets the x-axis at those points. The quadratic equations guide develops this roots-and-graph connection through factoring, the discriminant, and parabolas.
A polynomial function uses the same input-output reading you already know. The functions guide develops notation, graphs, and the vertical-line check that applies here.
Two graph rules follow straight from the factored form:
- End behavior rests only on the leading term aₙxⁿ. An even degree points both ends the same way; an odd degree points them opposite ways. The sign of aₙ sets up or down. The constant term does not enter it.
- Multiplicity sets how the graph meets an axis. A zero from an odd power, like (x - 1), crosses the axis; a zero from an even power, like (x + 3)², touches it and turns around.
A degree-n polynomial has at most n x-intercepts and at most n - 1 turning points. Read that as a ceiling, not an exact count, and remember that polynomial graphs are continuous and smooth, with no breaks or sharp corners.
Worked Examples and Checks
Example 1: Read degree and leading coefficient
Identify the degree, leading term, and leading coefficient of 5t⁵ - 2t³ + 7t.
The highest power is 5, so the degree is 5. The leading term is 5t⁵, and the leading coefficient is 5.
Check: the term 7t has degree 1, and no term beats 5, so the degree holds.
Example 2: Multiply and verify
Multiply (x - 4)(x + 4).
That is a difference of squares: x² - 4² = x² - 16.
Check by FOIL: x² + 4x - 4x - 16 = x² - 16. The middle terms cancel, and the product has degree 2, the sum of the two degree-1 factors.
Example 3: Factor and connect to a zero
Factor x² - 6x + 9.
This is a perfect-square trinomial: x² - 6x + 9 = (x - 3)².
Check: P(3) = 9 - 18 + 9 = 0, so 3 is a zero, matching the factor (x - 3). Because the factor is squared, the multiplicity is even, and the graph touches the axis at x = 3 instead of crossing it.
Common Polynomial Mistakes
- Reading degree as the number of terms. Degree is the highest power, not the term count.
- Reading end behavior from the constant term. End behavior comes only from the leading term.
- Assuming every zero crosses the axis. An even-multiplicity zero touches and turns around.
- Writing turning points as exactly n - 1. It is at most n - 1.
- Dropping the minus sign. In subtraction, spread the negative across every term of the second polynomial.
- Skipping the GCF. Factor out the greatest common factor before matching any structure method.
- Treating x⁻¹ or √x as polynomial terms. Negative and fractional exponents take the expression out of polynomial form.
Where Polynomials Fit in Algebra
Polynomials bring exponents and arithmetic together into reusable expressions. The Algebra 1 study guide shows how expressions, equations, and graphs build toward factoring and polynomial functions.
After the basics, factoring, division, and the full behavior of polynomial functions open up. The Algebra 2 guide maps how zeros, graphs, and rational expressions extend this same work. For a focused next step on polynomial fractions, use the rational expressions guide.
A compact review cycle works well:
- State the whole-number-exponent test in words.
- Read degree, leading term, and leading coefficient.
- Add, subtract, and multiply once, then combine like terms.
- Factor by GCF first, then match the structure.
- Multiply the factors back and check the signs.
How to Practice Polynomials with Gauth
No single lesson covers the whole path from definition through factoring to graphs. To focus on one weak point, open Gauth Math Courses, choose Create Course, and enter a topic such as "multiplying binomials and factoring quadratics" plus your level. Finish one activity, then return to an example here and work it without notes.
Sources and Review
OpenStax College Algebra 2e: Polynomials, accessed October 13, 2026.OpenStax College Algebra 2e: Factoring Polynomials, accessed October 13, 2026.OpenStax College Algebra 2e: Power Functions and Polynomial Functions, accessed October 13, 2026.OpenStax College Algebra 2e: Zeros of Polynomial Functions, accessed October 13, 2026.