A polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.
In Spanish
polinomio
Also called
polynomial expression, polynomial function (when written as f(x))
How to say it
pol-ee-NO-mee-ul
Grade level
Algebra 1
Algebra 1Algebra 2CCSS HSA-APR.A.1SATACT
Polynomial in a picture
A polynomial is made of terms joined by addition or subtraction, each a coefficient times a variable raised to a whole-number exponent. Division by a variable or a root of a variable rules an expression out.
Formal definition and notation
A polynomial in x is a sum of finitely many terms of the form axn, where each coefficienta is a number and each exponentn is a whole number (0, 1, 2, …). A polynomial with one term is a monomial, with two terms a binomial, with three terms a trinomial. The degree of a term is the sum of the exponents of its variables, and the degree of the polynomial is the highest degree of its terms. In standard form the terms are written in descending order of degree. Adding, subtracting or multiplying polynomials always gives another polynomial.
Symbol
Read it as
Meaning
P(x)
P of x
a name for a polynomial in the variable x, such as P(x) = x3 − 2x
A binomial is one kind of polynomial: exactly two terms, like x + 9. Polynomial is the whole family, with any number of terms.
Quick check: 3 questions
Question 1 of 3
Which expression is a polynomial?
Show answer and explanation
Answer: C. In 5x3 − 12x every exponent (3 and 1) is a whole number; a fraction as a coefficient is fine. Choice A divides by x, B takes a root of x, and D has a negative exponent.
Question 2 of 3
What is the degree of 6x2y3 − 4xy + 9?
Show answer and explanation
Answer: D. The degree of 6x2y3 is 2 + 3 = 5, the highest of all the terms. Choices A and B use only one exponent, and C is a coefficient.
Question 3 of 3
Simplify (4x2 + x − 6) − (x2 − 3x + 2).
Show answer and explanation
Answer: B. Subtracting changes every sign in the second polynomial: 4x2 + x − 6 − x2 + 3x − 2 = 3x2 + 4x − 8. Choice A forgets to change the signs, C adds the polynomials, and D changes only the middle sign.
Polynomial: frequently asked questions
Is a single number a polynomial?
Yes. A number such as 5 is a constant polynomial, a monomial of degree 0, because 5 = 5x0. A single variable like x is also a polynomial, of degree 1. The zero polynomial, 0, is a special case: most textbooks leave its degree undefined, because there is no nonzero term to take the highest exponent from.
Can a polynomial have fractions or decimals?
Yes, in the coefficients. 12x2 − 0.75x + 3 is a polynomial. What is not allowed is a variable in a denominator, like 3x, or a fraction or negative number in an exponent, like x12 or x−1. The rule is about the exponents of the variables, not about the numbers in front.
What is the difference between a polynomial, a monomial and a binomial?
They are names by number of terms. A monomial has one term (7x2), a binomial has two (x − 4), and a trinomial has three (x2 + 2x + 1). Polynomial is the general word for any of them, including expressions with four or more terms.
CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5MonomialA monomial is a single term made of a number, a variable, or a product of numbers and variables with whole-number exponents, such as 8, x, or −7x2y. It has no addition or subtraction, and no variable in a denominator. Its degree is the sum of the exponents on its variables.Algebra · Grade 8, Algebra 1BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2TrinomialA trinomial is a polynomial with exactly three terms, such as x2 + 5x + 6 or 2a − 3b + 7. The terms are joined by addition or subtraction and must be unlike terms, so nothing combines. The most common kind is the quadratic trinomial ax2 + bx + c, which students learn to factor into two binomial factors.Algebra · Algebra 1, Algebra 2DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7Standard FormStandard form of a number is the usual way of writing it with digits, such as 4,306, with each digit in its place-value position. It is the short form you get from expanded form (4,000 + 300 + 6) or from word form (four thousand, three hundred six). Zeros hold any empty places.Number System · Grade 2, Grade 3, Grade 4Leading CoefficientThe leading coefficient of a polynomial is the number multiplying the term with the highest power of the variable. In 3x2 − 5x4 + 2x, the highest power is x4, so the leading coefficient is −5. Write the polynomial in standard form first; its sign tells which way the graph’s ends point.Algebra · Algebra 1, Algebra 2Constant TermA constant term is a term in an expression or equation that has no variable, so its value never changes. In 3x2 − 5x + 7, the constant term is 7. Its sign belongs to it: in x2 + 4x − 9 it is −9. In a word problem it is usually a fixed amount, such as a one-time fee.Algebra · Grade 6, Grade 7, Algebra 1Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6ExpressionAn expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.Algebra · Grade 6, Grade 7, Grade 8EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2Degree of a PolynomialThe degree of a polynomial is the highest degree of any of its terms. For one variable, that is the largest exponent: 4x3 − 2x5 + 7x has degree 5. For a term with several variables, add their exponents, so 3x2y4 has degree 6. A nonzero constant such as 9 has degree 0.Algebra · Algebra 1, Algebra 2End BehaviorEnd behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x3 + x rises on the left and falls on the right.Functions · Algebra 2, Precalculus