Polynomial
The sum or difference of terms which
have variables raised to
positive integer powers and
which have coefficients that may be real or complex.
The following are all polynomials: 5x3 – 2x2 +
x – 13, x2y3 +
xy, and (1 + i)a2 + ib2.
Note: Even though the prefix poly- means many,
we use the word polynomial to refer to polynomials with 1 term
(monomials),
2 terms (binomials), 3 terms, (trinomials), etc.
Standard form for a polynomial in one variable:
anxn + an–1xn–1 + ··· + a2x2 + a1x + a0
See
also
Polynomial facts
Worked Example
Problem: Identify the degree, leading coefficient, and constant term of the polynomial 3x⁴ − 7x² + 5x − 2. Then rewrite it in standard form.
Step 1: Write the polynomial and check that it is already in standard form (terms ordered from highest power to lowest power of x).
3x4−7x2+5x−2 Step 2: Identify the degree. The highest exponent on x is 4, so this is a degree-4 polynomial (also called a quartic).
Degree=4 Step 3: Identify the leading coefficient. It is the coefficient of the highest-degree term, which is 3.
Leading coefficient=3 Step 4: Identify the constant term. This is the term with no variable attached, which is −2.
Step 5: Count the terms. There are four terms: 3x⁴, −7x², 5x, and −2. Notice that the x³ term is missing — its coefficient is simply 0.
3x4+0x3−7x2+5x−2 Answer: The polynomial 3x⁴ − 7x² + 5x − 2 has degree 4, leading coefficient 3, constant term −2, and four terms. It is already in standard form.
Another Example
Problem: Add the polynomials (2x³ + 4x − 6) and (x³ − 3x² + 5).
Step 1: Write both polynomials and align like terms (terms with the same power of x).
(2x3+0x2+4x−6)+(x3−3x2+0x+5) Step 2: Combine like terms by adding their coefficients.
(2+1)x3+(0−3)x2+(4+0)x+(−6+5) Step 3: Simplify each group.
3x3−3x2+4x−1 Answer: The sum is 3x³ − 3x² + 4x − 1, which is a degree-3 polynomial with four terms.
Frequently Asked Questions
Is x⁻² + 3 a polynomial?
No. A polynomial requires all variable exponents to be non-negative integers (0, 1, 2, 3, …). The term x⁻² has a negative exponent, so the expression is not a polynomial. Similarly, √x (which equals x^(1/2)) and 1/x (which equals x⁻¹) disqualify an expression from being a polynomial.
Can a single number like 7 be a polynomial?
Yes. The number 7 is a polynomial of degree 0, called a constant polynomial. You can think of it as 7x⁰. Even 0 by itself is considered a polynomial, though by convention the zero polynomial has no defined degree.
Polynomial vs. Rational Expression
A polynomial is a sum of terms with non-negative integer exponents on the variables. A rational expression is a fraction where both the numerator and denominator are polynomials, such as (x² + 1)/(x − 3). Every polynomial can be written as a rational expression (put it over 1), but not every rational expression is a polynomial — dividing by a variable disqualifies it.
Why It Matters
Polynomials are one of the most fundamental objects in all of mathematics. They model real-world relationships — from the parabolic path of a thrown ball (degree 2) to profit functions in business. Because polynomials are smooth, continuous, and easy to differentiate and integrate, they serve as building blocks in calculus, engineering, computer graphics, and data science.
Common Mistakes
Mistake: Thinking that expressions with negative or fractional exponents on a variable (like x⁻¹ or x^(1/2)) are polynomials.
Correction: Every exponent on a variable in a polynomial must be a non-negative integer (0, 1, 2, 3, …). Terms like 1/x or √x violate this rule.
Mistake: Confusing the degree of a polynomial with the number of terms.
Correction: The degree is the highest exponent (e.g., x⁵ + 1 has degree 5), while the number of terms is how many separate parts are added or subtracted (x⁵ + 1 has 2 terms, making it a binomial).
