Special Product Formulas

Quick Reference

#	Formula	Name
1		Square of a Sum
2		Square of a Difference
3		Cube of a Sum
4		Cube of a Difference
6		Product of Binomials with a Common Term
7		Difference of Squares
8		Difference of Cubes
9		Sum of Cubes

The Formulas

The following special formulas are vastly used in algebra and calculus, and should be memorized. You can verify each of the formulas by actual multiplication.

Here and represent real numbers, variables, or algebraic expressions.

Squares of Binomials

Cubes of Binomials

Note that the Square of a Difference formula can be obtained by replacing with in the Square of a Sum formula. Similarly, replacing with in the Cube of a Sum formula yields the Cube of a Difference formula.

Binomial Expansion

The formulas above handle the cases and . For higher powers, the coefficients of the expansion of can be read directly from the Pascal-Newton-Khayyam triangle (commonly called Pascal's triangle). Each row of the triangle corresponds to one value of , and the entries in that row are the coefficients of in order.

	Coefficients of
0
1
2
3
4
5
6

Each entry is the sum of the number directly above it and the number to the left of that one. For example, in row 4 the entry equals (the directly above it and the to its left in row 3), and each equals or by the same rule. The first and last entries of every row are always . To expand , read off row , attach the corresponding powers of (descending from to ) and (ascending from to ), and write out the sum.

For example, using row :

And using row :

Further Study: The Binomial Coefficient Formula

The general formula for any power

For large , reading off the triangle row by row can be tedious. The entry in row at position (counting from ) is given by the **binomial coefficient** where (with by convention). This gives the general expansion: and You can verify that the binomial coefficients recover the triangle entries: , , , , , which matches row above.

Other Product Formulas

Trinomial Formulas

The square and cube of a trinomial can be derived from the binomial formulas by treating two of the three terms as a single unit. Set , so that , and then apply the binomial formulas to the pair .

Square of a trinomial. Apply the Square of a Sum formula to :

In other words, the square of a trinomial equals the sum of the squares of each term plus twice the product of every pair of distinct terms: , , from squaring each term, and , , from doubling each pairwise product.

Derivation of the cube of a trinomial

Apply Formula 3 to , where : Expand each term using the binomial formulas: Adding all terms: In other words, the cube of a trinomial equals the sum of the cubes of each term (, , ), plus three times the square of each term multiplied by each of the other two terms (, , , , , ), plus six times the product of all three terms ().

Common Mistake: Squaring a Binomial

Worked Examples

Frequently Asked Questions

What are special product formulas?

Special product formulas are identities that give the expanded form of certain polynomial products directly, without performing full term-by-term multiplication. Because the same patterns (squares of binomials, difference of squares, cubes) appear constantly in algebra and calculus, memorizing these formulas saves significant time.

What is the difference of squares formula?

The difference of squares formula states that . It applies whenever two identical expressions are multiplied with opposite signs between their terms. For example, .

What is a perfect square trinomial?

A perfect square trinomial is a trinomial of the form or , which factors as or respectively. For example, and .

What is the most common mistake when squaring a binomial?

The most common error is writing , which omits the middle term . The correct formula is . Similarly, , not .

What is the binomial theorem?

The binomial theorem gives the full expansion of for any positive integer : where is the binomial coefficient (read " choose "). The formulas for and are the special cases and , and the triangle rows give the coefficients for all small at a glance.