Binomial — Definition, Examples, Operations

TL;DR

A binomial is an algebraic expression with exactly two unlike terms — like 3x + 5 or x² − 9. This article covers the definition, three worked examples at Quick/Standard/Stretch tiers, the four basic operations on binomials (with the FOIL method for multiplication), and the link to the binomial theorem for higher powers.

Two Terms, Connected by a Sign

Most useful algebra happens at the binomial level. Linear equations are binomial = constant. Quadratics factor into products of binomials. Probability uses binomial coefficients. The binomial is small enough to manipulate by hand, big enough to model real situations.

The word "binomial" combines bi- (two) with nomial (term). It is exactly what the etymology says: a two-term expression. Every later operation — adding, subtracting, multiplying, raising to a power — starts from this definition.

What a Binomial Is

A binomial is an algebraic expression with exactly two unlike terms, connected by an addition or subtraction sign. The two terms can include variables, exponents, and coefficients, but they cannot be like terms (which would combine into a monomial).

Examples — 3x + 5, x² − 9, 4a³b − 7b², 2x + y, 12x⁴ − 3.

What is not a binomial:

• 5x — one term (monomial).
• 3x + 2x — two like terms that combine to 5x (monomial after simplification).
• x² + 3x + 5 — three terms (trinomial).
• x + 2√x — contains a fractional exponent, so often classified separately from polynomial binomials.

Quick facts.

• Term count: exactly 2.
• Terms: must be unlike — they cannot combine.
• Connector: + or −.
• Degree: the highest degree of either term.
• Special case: if both terms are perfect squares with opposite signs, the binomial is a difference of squares: a² − b² = (a + b)(a − b).

The Four Operations on Binomials

Addition

Add binomials by combining like terms.

(3x + 5) + (2x + 4) = 5x + 9.

Subtraction

Distribute the negative across the second binomial, then combine like terms.

(3x + 5) − (2x + 4) = 3x + 5 − 2x − 4 = x + 1.

Multiplication — The FOIL Method

To multiply two binomials, multiply each term of the first by each term of the second. The mnemonic FOIL — F irst, O uter, I nner, L ast — names the four products in order.

$$(a + b)(c + d) = \underbrace{ac}{\text{First}} + \underbrace{ad}{\text{Outer}} + \underbrace{bc}{\text{Inner}} + \underbrace{bd}{\text{Last}}.$$

Example. (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15.

Division

Long division or synthetic division. Division by a binomial reduces the degree of the dividend by 1 if the division is clean.

Example. (x² + 5x + 6) ÷ (x + 2) = x + 3.

Worked Examples of Binomial

Quick. Multiply (x + 4)(x + 2).

FOIL: First x⋅x = x², Outer x⋅2 = 2x, Inner 4⋅x = 4x, Last 4⋅2 = 8.

x² + 2x + 4x + 8 = x² + 6x + 8.

Standard (Wrong Path First — The Mistake Worth Making Once). Multiply (2x − 3)(x + 5).

The wrong path. The rusher distributes only the 2x across the second binomial: (2x − 3)(x + 5) = 2x² + 10x − 3.

Check at x = 1: (2 − 3)(1 + 5) = (−1)(6) = −6. The wrong expansion at x = 1 is incorrect; the values disagree.

The rescue. Apply FOIL — every term of the first binomial multiplies every term of the second.

(2x − 3)(x + 5) = 2x⋅x + 2x⋅5 + (−3)⋅x + (−3)⋅5 = 2x² + 10x − 3x − 15 = 2x² + 7x − 15.

Stretch. Use the binomial theorem to expand (x + 2)⁴.

Read row 4 of Pascal's triangle: 1, 4, 6, 4, 1.

(x + 2)⁴ = x⁴ + 4x³(2) + 6x²(2²) + 4x(2³) + 2⁴ = x⁴ + 8x³ + 24x² + 32x + 16.

Where Binomials Show Up in the Real World

The binomial is the entry point for nearly every quantitative model.

• Compound interest: A = P(1 + r)ⁱ — the principal multiplies by the binomial (1 + r) each year.
• Probability: The binomial distribution gives the chance of k successes in n independent trials.
• Genetics: Mendelian inheritance models allele frequencies as (p + q)² in the Hardy-Weinberg equation.
• Physics linearization: For small h, (1 + h)ⁿ ≈ 1 + nh.
• Computer arithmetic: Multiplication of binary numbers uses binomial multiplication.

The Binomials Errors That Cost Most Marks

1. Treating like terms as if they were unlike.

Count terms without simplifying first.

2. Forgetting to distribute the second term in FOIL.

Every term of the first binomial multiplies every term of the second.

3. Sign errors in subtraction.

The minus distributes across every term.

4. Confusing (a+b)² with a² + b².

The correct way is to include the middle term.

The Mathematicians Who Shaped Binomial Algebra

• Pingala (c. 200 BCE, India): described the binomial coefficients in his Chandaḥśāstra.
• Omar Khayyam (1048–1131, Persia): worked out integer-power expansions of binomials.
• Isaac Newton (1643–1727, England): extended the binomial theorem to any real power in 1665.

Conclusion

• A binomial is an algebraic expression with exactly two unlike terms.
• Add and subtract by combining like terms; multiply by FOIL; divide by long division.
• The single most common mistake is forgetting to distribute the second term across both pieces of the other binomial.
• (a+b)² = a² + 2ab + b² — never drop the middle term.
• The binomial theorem extends multiplication of binomials to any positive integer power.

Practice These Three Before Moving On

1. Multiply (x + 7)(x − 3).
2. Expand (2x + 5)².
3. Factor the difference of squares x² − 25.