1
The exponents are whole numbers
0, 1, 2, 3, … and nothing else. That single restriction is what makes the whole course work: it is why like terms exist, why degree behaves predictably, and why every technique here terminates.
Lesson · Polynomials and Factoring · Polynomial arithmetic
Polynomials, Degree and Standard Form
A polynomial is a finite sum of terms, each a number times a whole-number power of the variable. Establish which expressions qualify, why 1/x and √x do not, and what degree measures once they do.
3x⁴ − 5x² + x − 7 polynomial: degree 4, leading coefficient 3
2/x = 2x⁻¹ NOT a polynomial (negative exponent)
√x = x^(1/2) NOT a polynomial (fractional exponent)
2ˣ NOT a polynomial (variable in the exponent)
Four expressions, one polynomial
Core ideas
What this lesson establishes
Every rule in this course is stated for polynomials and is guaranteed only for polynomials. So the first job is to say exactly which expressions those are.
2
Degree is the handle on the whole expression
One number, read off the highest power, that predicts how the expression behaves under addition and multiplication and how many roots it can have. Most decisions in this course start by looking at it.
3
Standard form is a convention, not a rule of algebra
Writing terms in descending degree changes nothing about the expression. It is worth doing because every later procedure — division in “Polynomial Long Division” especially — assumes the terms are in that order.
The material
What a polynomial is
One definition, the three ways an expression fails it, and the vocabulary that follows.
Definition · Polynomial in one variable
A polynomial in x is an expression that can be written as a finite sum aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where n is a whole number and the coefficients aₙ, …, a₀ are numbers.
"Can be written as" matters: x(x + 1) is a polynomial, because multiplying it out gives x² + x.
Each aₖxᵏ is a term. The number aₖ is the coefficient of that term, and a₀ — the term with no x — is the constant term.
Three kinds of expression fail the definition, each for the same underlying reason: the exponent is not a whole number.
- Negative exponents. 2/x is 2x⁻¹, and −1 is not a whole number. Expressions like this are rational, and Rational and Radical Expressions handles them.
- Fractional exponents. √x is x^(1/2). This is a radical expression, also Rational and Radical Expressions.
- The variable in the exponent. 2ˣ has a constant base and a variable exponent, which is the exact reverse of x². That is an exponential, and Exponential and Logarithmic Functions is about it.
A polynomial may contain more than one variable: 4x²y − 3xy + y³ is a polynomial in x and y. The same restriction applies to every variable in every term.
Definition · Degree, leading coefficient, standard form
The degree of a polynomial is the largest exponent appearing on the variable, once like terms are combined. The coefficient of that term is the leading coefficient. A polynomial is in standard form when its terms are written in descending order of degree.
For a term in several variables, the degree of that term is the sum of the exponents: 4x²y has degree 3.
"Once like terms are combined" is not a decoration. In x³ + 5x − x³ the highest power written is 3, but the x³ terms cancel and the polynomial is 5x, of degree 1. Degree is a property of the polynomial, not of the way it happens to be written.
Two cases at the bottom of the scale. A nonzero constant such as 7 is 7x⁰, so it has degree 0. The zero polynomial has no degree at all — there is no highest power to point at, and every convention for assigning it one exists only to keep later statements tidy.
Short polynomials are named by their number of terms: one term is a monomial, two a binomial, three a trinomial. Beyond three, nobody bothers.
Example · Reading a polynomial
−2x³ + 5x + 8 is a trinomial in standard form. Degree 3, leading coefficient −2, constant term 8. The coefficient of x² is 0, which is worth saying out loud — “Polynomial Long Division” will make you write that zero down.
Interactive
What a polynomial is
Degree, terms, and the two things that disqualify an expression
Method
Putting an expression in standard form
Four moves, in this order. The first one can end the exercise.
1
Check that it is a polynomial at all
Look at every exponent on every variable. A negative one, a fractional one, or a variable sitting in an exponent means the techniques in this course do not apply, and a different course does.
2
Multiply out anything in brackets
The definition asks whether the expression can be written as a sum of terms, so x(x + 1) qualifies. You cannot read the degree off a factored form reliably until “Multiplying Polynomials”.
3
Combine like terms
Terms with the same power of the same variable add together. Do this before reading the degree, because cancellation can lower it.
4
Order the terms downward, then read off the data
Highest power first. The degree and the leading coefficient are then the first thing you see, which is the whole point of the convention.
Polynomial or not
Score 0 / 0Choose the answer you can justify.
Classify, then standardise
Five expressions. Say whether each is a polynomial, and give the reason when it is not.
1. 5x − 3x³ + 8 + x³ polynomial
combine: −3x³ + x³ = −2x³
standard: −2x³ + 5x + 8 degree 3, leading coefficient −2
2. 4x²y − 3xy + y³ polynomial in x and y, degree 3
(4x²y has degree 2 + 1 = 3; y³ has degree 3)
3. x² + 3/x NOT a polynomial: 3/x = 3x⁻¹
4. √x + 1 NOT a polynomial: √x = x^(1/2)
5. 7 polynomial, degree 0 (it is 7x⁰)
Line 1 is the one that catches people. Written as it stands, the highest power on the page is 3 and the leading coefficient looks like −3. Neither reading survives combining like terms, and combining like terms is not optional — it is how you find out what the polynomial actually is.
For a faded pass, start 6 − 4x³ + 2x − x³ with the supplied decision: every exponent is a whole number, so it is a polynomial. Combine and order the terms yourself, then name the degree, leading coefficient and term count. Check against −5x³ + 2x + 6: degree 3, leading coefficient −5, three terms. If you wrote −3x³, you combined coefficients as though the second cubic term were positive.
Common mistakes
1
Reading the degree off the first term writtenStandard form makes that reliable; nothing else does. 5x − 3x³ + 8 + x³ has degree 3 with leading coefficient −2, and you can only know that after combining.
2
Thinking a fractional coefficient disqualifies an expression(1/2)x³ − 0.7x is a perfectly good polynomial. The definition restricts the exponents; the coefficients can be any numbers.
3
Calling a factored expression "not a polynomial"(x + 1)(x − 4) is a polynomial, because it can be written as x² − 3x − 4. The definition asks what the expression equals, not how it is currently printed.
Check your understanding
✓
Finish when you can reject a non-polynomial and name the exponent that spoils it.
Given a mixed list you should sort it without hesitating, and for each rejection point at the specific exponent — negative, fractional, or variable — that fails the definition. That habit is what stops you applying a factoring technique to 1/x + x in Rational and Radical Expressions and getting nonsense.
Note: The lab reports the degree after combining, which is why it is worth typing in something that cancels. Degree is the number this whole course steers by: it bounds how many roots a polynomial can have (“Graphs of Polynomial Functions”) and it tells you when polynomial division has finished (“Polynomial Long Division”).