Polynomial
In short: A polynomial is an expression made of terms added or subtracted, where each term pairs a coefficient with a variable to a counting-number power.
Algebra 1AlgebraA-APR.A.1
What is Polynomial?
Take pieces like \(4x^3\), \(-7x^2\), \(x\) and \(-9\), join them with plus and minus signs, and you have a polynomial. Every piece is a term: a coefficient times a power of the variable, where the exponent is 0, 1, 2, 3 or another counting number.
That rule excludes negative exponents, fractional exponents, variables in denominators and variables under radicals. Such expressions can be added, subtracted and multiplied, and the outcome stays in the family, which is why they turn up in formulas for area, volume and motion.
How to spot it
- An expression has several terms joined by plus and minus signs, each with a whole-number exponent on x.
- A question asks for the degree, the leading coefficient or the number of terms.
- You are told to add, subtract or multiply expressions such as \(3x^2 + 5x – 1\).
- Words like monomial, binomial or trinomial are used to name expressions by their number of terms.
The method
- Split the expression into terms at each plus or minus sign, keeping each sign with its term.
- Check every term: the variable’s exponent must be a whole number and the variable must not sit in a denominator or under a root.
- Write the terms in descending order of exponent to reach standard form.
- Name the degree (largest exponent) and the leading coefficient (number on that term).
Polynomial: solved example
Problem
Decide which of these are polynomials: \(5x^2 – 3x + 1\), \(\frac{4}{x} + 2\), \(\sqrt{x} + 6\), and \(7\).
Solution
\(5x^2 – 3x + 1\): exponents 2, 1, 0, all whole numbers, so it is a polynomial.
\(\frac{4}{x} + 2\) has x in a denominator, which is \(x^{-1}\), and \(\sqrt{x} + 6\) uses \(x^{1/2}\); neither is allowed. The number 7 is a constant term of degree 0. The ones that qualify are \(5x^2 – 3x + 1\) and \(7\).
Try it
Question 1
Write \(3x – 5x^3 + 2\) in standard form.
Show the answer
Order by decreasing exponent: \(-5x^3 + 3x + 2\).
Question 2
Is \(x^{-2} + 1\) a polynomial?
Show the answer
No. The exponent \(-2\) is negative, not a whole number.
Polynomial vs. Algebraic Expression
Every polynomial is an algebraic expression, but not the other way around. An algebraic expression may include division by a variable, square roots of a variable or other operations. A polynomial is the restricted kind, built only from whole-number powers combined with addition, subtraction and multiplication.
Keep going
Lessons
Worksheets and quizzes
- Adding, Subtracting and Multiplying Polynomials – Worksheet
- Adding, Subtracting and Multiplying Polynomials – Quiz
- Structure of Algebraic Expressions – Worksheet
- Structure of Algebraic Expressions – Quiz
Related words
Whole dictionary: A–Z · Algebra 1 · Algebra