Key takeaway: A polynomial is an algebraic expression that is a sum of terms, each a number times variables raised to whole-number exponents, with no variables in denominators or under roots.
Grade 9 – Algebra 1A-APR.A.1Expressions and Equations
Definition of Polynomial
A polynomial is built by adding or subtracting terms such as \(4x^{3}\), \(-7x^{2}\), \(2x\) and \(-15\). Every term is a number (the coefficient) multiplied by variables with exponents that are whole numbers: 0, 1, 2, 3 and so on.
Some expressions break the rules. A variable in a denominator, such as \(\frac{3}{x}\), a negative exponent like \(x^{-2}\), a fractional exponent or a variable under a radical sign all disqualify an expression. A single number such as 8 qualifies, and so does a lone term like \(5y\). These expressions are the main objects of Algebra 1 because they can be added, subtracted, multiplied and factored.
Formula and notation
Terms are usually written in descending order of exponents. In one variable:
\[a_n x^{n} + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0\]
The first coefficient \(a_n\) is the leading coefficient, \(a_0\) is the constant term, and the exponent \(n\) of the highest-power term is the degree of the polynomial.
Key facts
- Each exponent in a polynomial is 0 or a positive whole number.
- Adding, subtracting or multiplying two of them always gives another.
- Special names: monomial (one term), binomial (two terms), trinomial (three terms).
- Terms with the same variable part are like terms and can be combined.
Polynomial example
Which of these are polynomials? (a) \(6x^{2} – 4x + 11\), (b) \(3x^{-2} + 5\), (c) \(\sqrt{x} + 1\), (d) \(8\).
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(a) The exponents are 2, 1 and 0, all whole numbers, so it qualifies.
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(b) The exponent \(-2\) is negative, and (c) \(\sqrt{x} = x^{1/2}\) has a fractional exponent.
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(d) The number 8 is \(8x^{0}\), a constant term with exponent 0.
(a) and (d) qualify; (b) and (c) do not.
Common mistake
Watch out
Cleo says that \(\frac{x^{2} + 1}{4}\) does not qualify because it contains a fraction bar.
Why it is wrong
The test is about the variable, not the fraction bar. Dividing by the number 4 is the same as multiplying by \(\frac{1}{4}\), so the expression equals \(\frac{1}{4}x^{2} + \frac{1}{4}\), whose exponents are 2 and 0. It qualifies. An expression like \(\frac{4}{x^{2} + 1}\) is not, since the variable is in the denominator.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 1 vocabulary · Expressions and Equations