Key takeaway: The degree of a polynomial is the greatest degree among its terms, where a term’s degree is the sum of the exponents on its variables.
Grade 9 – Algebra 1A-APR.A.1Expressions and Equations
Definition of Degree of a Polynomial
The degree of a polynomial tells how “powerful” its highest term is. Combine like terms first, then look at each remaining term. In one variable, the degree is simply the largest exponent: \(7x^{2} + 3x^{5} – x\) has degree 5.
With several variables, add the exponents inside each term, then take the greatest sum. A constant such as 9 has degree 0, and the number 0 alone has no defined degree. The degree gives names to polynomials (linear, quadratic, cubic), predicts the shape of a graph and the greatest number of solutions an equation can have.
Formula and notation
For a term \(c\,x^{a}y^{b}\), the term degree is \(a + b\). The degree of a polynomial is
\[\deg(P) = \max\{\text{degree of each term}\}\]
Common names by degree: 0 constant, 1 linear, 2 quadratic, 3 cubic, 4 quartic, 5 quintic.
Key facts
- Combine like terms before finding the degree, since terms can cancel.
- The degree can be located in any term, not necessarily the first one written.
- A nonzero constant has degree 0.
- A polynomial of degree n has at most n real zeros.
Degree of a Polynomial example
Find the degree of \(3x^{2}y^{3} – 5x^{4} + 2xy + 8\) and name the leading term.
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Add exponents in each term: \(3x^{2}y^{3}\) gives \(2 + 3 = 5\); \(-5x^{4}\) gives 4.
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\(2xy\) gives \(1 + 1 = 2\), and the constant 8 gives 0.
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The greatest of 5, 4, 2 and 0 is 5.
The degree is 5, and the leading term is \(3x^{2}y^{3}\).
Common mistake
Watch out
Lena says the degree of \(7x^{2} + 3x^{5} – x\) is 2 because \(7x^{2}\) comes first.
Why it is wrong
The order of writing does not matter. The exponents are 2, 5 and 1, and the largest is 5, so the degree is 5. Rewriting in descending order, \(3x^{5} + 7x^{2} – x\), puts the highest-degree term first and avoids the slip.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 1 vocabulary · Expressions and Equations