A 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.
A binomial has exactly two terms. Multiplying two binomials gives four partial products, and combining the two like middle terms usually leaves a trinomial.
Formal definition and notation
A binomial is a polynomial that is the sum or difference of two unlike monomials, such as axm + bxn with a ≠ 0, b ≠ 0 and m ≠ n. Each monomial is a term: a number, a variable, or a product of a number and variables with whole-number exponents. The terms must be unlike, because like terms combine into a single term.
Symbol
Read it as
Meaning
a + b
a plus b
the simplest binomial: two terms joined by addition
(x + 3)(x − 3)
the quantity x plus 3 times the quantity x minus 3
a product of two binomials; parentheses keep each binomial together
A trinomial has three terms, such as x2 + 5x + 6; a binomial has two, such as x + 6.
Binomialvs.Binomial distribution
In probability, a binomial distribution counts successes in repeated yes/no trials, like heads in 10 coin flips. It shares the ‘two’ idea (two outcomes) but is not an algebraic expression.
Polynomial is the whole family: any number of terms. A binomial is the polynomial with exactly two terms, so every binomial is a polynomial but x2 + x + 1 is not a binomial.
Quick check: 3 questions
Question 1 of 3
Which expression is a binomial?
Show answer and explanation
Answer: C.5a − 2b has two unlike terms. 7x is one term, x2 + 3x + 2 has three, and 4x + 3x simplifies to 7x, a single term.
Question 2 of 3
What is (x + 5)(x + 2)?
Show answer and explanation
Answer: A. FOIL gives x2 + 2x + 5x + 10 = x2 + 7x + 10. Choice B skips the outer and inner products, choice C adds 5 and 2 instead of multiplying them, and choice D adds the binomials.
Question 3 of 3
Is 3x + 5x a binomial?
Show answer and explanation
Answer: B. Like terms combine: 3x + 5x = 8x, a monomial. Choice D is wrong because x + 5 is a binomial with only one variable.
Binomial: frequently asked questions
What are examples of binomials?
x + 1, 2y − 7, a2 − b2, 4x3 + 9x and m + n are all binomials: each has exactly two unlike terms. x2 − 9 is a special binomial called a difference of squares, because it factors as (x + 3)(x − 3). 5x is not a binomial (one term), and neither is x2 + x + 1 (three terms).
How do you multiply two binomials?
Multiply each term of the first binomial by each term of the second, then combine like terms. FOIL is a memory aid for the order: First, Outer, Inner, Last. For (x + 2)(x + 6): x2 + 6x + 2x + 12 = x2 + 8x + 12. An area model with a 2 by 2 grid of boxes does the same four products.
What is the difference between a monomial, binomial and trinomial?
The prefix counts the terms: mono means one, bi means two, tri means three. 7x2 is a monomial, 7x2 − 3 is a binomial and 7x2 − 3x + 1 is a trinomial. With four or more terms we usually just say polynomial. Always combine like terms first, because 2x + 3x counts as one term.
PolynomialA 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.Algebra · Algebra 1, Algebra 2DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3MonomialA 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 1NumberA 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 8Like 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 8AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3Difference of SquaresA difference of squares is an expression of the form a2 − b2: one perfect square subtracted from another. It always factors as (a + b)(a − b). For example, x2 − 49 = (x + 7)(x − 7). A sum of squares, a2 + b2, does not factor over the real numbers.Algebra · Algebra 1, Algebra 2FactoringFactoring 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 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 8TrinomialA 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 2SimplifySimplify 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 6OutcomeAn outcome is one possible result of a chance experiment, such as rolling a 4 on a number cube or getting tails on a coin flip. The set of all outcomes is the sample space, and a group of outcomes is an event. When outcomes are equally likely, each one has probability 1 divided by the number of outcomes.Probability · Grade 7, Grade 8, StatisticsArea ModelAn area model is a rectangle used to multiply or divide, with each side broken into place-value parts. To find 23 × 14, split the sides into 20 + 3 and 10 + 4, find the area of each smaller box (200, 30, 80, 12) and add these partial products: 322.Operations · Grade 3, Grade 4, Grade 5ExpressionAn 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 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 2FOIL MethodThe FOIL method is a memory aid for multiplying two binomials: multiply the First terms, the Outer terms, the Inner terms and the Last terms, then combine like terms. For (x + 2)(x + 5): x2 + 5x + 2x + 10 = x2 + 7x + 10. It is the distributive property applied twice.Algebra · Algebra 1