Binomial
In short: A binomial is a polynomial with exactly two terms, such as 3x − 8 or a² + 5, joined by a plus or minus sign.
Algebra 1AlgebraA-APR.A.1
What is Binomial?
The prefix bi- means two, and a binomial is a polynomial made of exactly two unlike terms. Expressions such as \(x + 4\), \(3y – 8\) and \(a^2 + 5\) qualify; \(7x\) alone is a monomial, and \(x^2 + 2x + 1\) is a trinomial.
Were the two terms alike they would merge into one, so the terms differ in their variable part. These two-term factors appear whenever you factor or multiply, as in \((x + 3)(x – 3)\).
How to spot it
- An expression has exactly two terms separated by one plus or minus sign.
- A product is written as two parentheses, each holding a sum or difference.
- A problem asks you to square something like \((x + 3)\), or to multiply it by another.
- Factoring a quadratic ends in two such factors.
The method
- To add or subtract, remove parentheses (distribute the minus sign), then combine like terms.
- To multiply two of them, use FOIL or the area model so that all four products appear.
- To square one, write it as a product of itself: \((a + b)^2 = a^2 + 2ab + b^2\).
- Finish by collecting like terms and writing the result in descending order.
Binomial: solved example
Problem
Subtract the binomials: \((9y + 4) – (2y – 6)\).
Solution
Distribute the minus sign to both terms of the second binomial: \(9y + 4 – 2y + 6\).
Combine like terms: \(9y – 2y = 7y\) and \(4 + 6 = 10\). The difference is \(7y + 10\).
Try it
Question 1
Does \(x^2 + 3x – 1\) have exactly two terms, and if not, what is the right name for it?
Show the answer
No. It has three terms, so it is a trinomial.
Question 2
Expand \((x + 3)^2\) and say why a middle term shows up.
Show the answer
\(x^2 + 3x + 3x + 9 = x^2 + 6x + 9\). The two cross products, 3x and 3x, combine into the middle term 6x.
Binomial vs. Monomial
A monomial is one term and a binomial is two, and the signs make the count visible. The classic trap is squaring: \((x + 3)^2\) is not \(x^2 + 9\), because the cross terms \(3x + 3x\) must be included, giving a trinomial.
Keep going
Lessons
Worksheets and quizzes
- Adding, Subtracting and Multiplying Polynomials – Worksheet
- Adding, Subtracting and Multiplying Polynomials – Quiz
Related words
Whole dictionary: A–Z · Algebra 1 · Algebra