# 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.