Multiplication of Algebraic Expressions: Rules & Examples

TL;DR

Multiplying algebraic expressions means using the distributive property to multiply every term of one expression by every term of the other, then combining like terms. This article covers the sign and exponent rules, monomial and binomial and polynomial products, the FOIL shortcut, worked examples, and the mistakes that cost the most marks.

What Is Multiplication of Algebraic Expressions?

Multiplication of algebraic expressions is the process of finding the product of two or more expressions by multiplying each term of one by each term of the other and combining like terms. An algebraic expression is a combination of variables, constants, and operations — for example, 3a+5b or 2x^2 - 4x + 7. Multiplying them rests on one idea, the distributive property: a(b+c) = ab + ac.

Two rules govern every product before you even start:

Sign rule: like signs give a positive product (+ × + = +, − × − = +); unlike signs give a negative product (+ × − = −).
Exponent rule: multiply powers of the same base by adding exponents, a^m × a^n = a^{m+n}.

The Four Types of Products

Which method you use depends on how many terms each expression has. A monomial has one term (5x), a binomial has two (x + 3), and a polynomial has two or more.

Monomial × monomial — multiply coefficients, add exponents of like variables: (6ab)(−3a^2b^3) = −18a^3b^4.
Monomial × polynomial — distribute the monomial across every term of the polynomial.
Binomial × binomial — use the distributive property, often remembered as FOIL (First, Outer, Inner, Last).
Polynomial × polynomial — multiply each term of the first by each term of the second, then combine like terms.

A short list of standard identities speeds up common binomial products:

(a+b)^2 = a^2 + 2ab + b^2
(a−b)^2 = a^2 - 2ab + b^2
(a+b)(a−b) = a^2 - b^2

You can always fall back on the distributive property, but recognising these three algebraic identities saves time.

The Box and Vertical Methods

FOIL and plain distribution work term by term, but two layouts make longer products easier to organise without dropping a term.

Box (area) method: draw a grid with the terms of one expression along the top and the terms of the other down the side. Fill each cell with the product of its row and column term, then add the cells and combine like terms.
Vertical (column) method: write one expression under the other, exactly like long multiplication of numbers. Multiply the top expression by each term of the bottom one, line up like terms in columns, and add. This is the tidiest layout for a polynomial times a polynomial.

Both methods give the same result as the distributive property; they only change how you keep track of the terms.

Properties of Multiplication of Algebraic Expressions

Multiplying algebraic expressions obeys the same structural laws as multiplying numbers.

Closure: the product of two algebraic expressions is again an algebraic expression.
Commutative: the order of the factors does not change the product, A × B = B × A.
Associative: grouping does not change the product, (A × B) × C = A × (B × C).
Distributive: multiplication distributes over addition, A(B + C) = AB + AC.
Multiplicative identity: multiplying any expression by 1 leaves it unchanged.

Examples of Multiplication of Algebraic Expressions

Example 1

Multiply the monomials 6ab and −3a^2b^3.

Multiply the coefficients:

6 × (−3) = −18.

Add exponents of like bases: a^1 × a^2 = a^3, b^1 × b^3 = b^4.

Combine:

6ab × (−3a^2b^3) = −18a^3b^4.

Final answer: −18a^3b^4

Example 2

Multiply the monomial 5a^2b^2 by the polynomial 3a^2 − 4ab + 6b^2.

Distribute the monomial across every term:

5a^2b^2 × 3a^2 = 15a^4b^2, 5a^2b^2 × (−4ab) = −20a^3b^3, 5a^2b^2 × 6b^2 = 30a^2b^4.

Combine:

5a^2b^2(3a^2 − 4ab + 6b^2) = 15a^4b^2 − 20a^3b^3 + 30a^2b^4.

Final answer: 15a^4b^2 − 20a^3b^3 + 30a^2b^4

Example 3

Multiply the binomials (3a+5b) and (5a−7b).

Use FOIL:

First: 3a × 5a = 15a^2
Outer: 3a × (−7b) = −21ab
Inner: 5b × 5a = 25ab
Last: 5b × (−7b) = −35b^2

Combine the like middle terms:

−21ab + 25ab = 4ab.

(3a+5b)(5a−7b) = 15a^2 + 4ab − 35b^2.

Final answer: 15a^2 + 4ab − 35b^2

Example 4

Expand (2x+3)^2 using an identity.

Match the pattern (a+b)^2 = a^2 + 2ab + b^2 with a=2x, b=3:

(2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9.

Final answer: 4x^2 + 12x + 9

Example 5

Multiply the polynomials (5x^2−6x+9) and (2x−3).

Multiply each term of the first by each term of the second:

2x(5x^2−6x+9) = 10x^3−12x^2+18x, −3(5x^2−6x+9) = −15x^2 + 18x − 27.

Add and combine like terms:

10x^3−12x^2−15x^2 + 18x + 18x − 27 = 10x^3−27x^2 + 36x − 27.

Final answer: 10x^3 − 27x^2 + 36x − 27

Example 6

Multiply (x+2)(x−5)(x+1), a product of three binomials.

Multiply the first two:

(x+2)(x−5) = x^2 − 3x − 10.

Now multiply that trinomial by the third binomial:

(x^2 − 3x − 10)(x+1) = x^3 − 2x^2 − 13x − 10.

Final answer: x^3 − 2x^2 − 13x − 10

Why Multiplying Expressions Matters: "Every term is a piece that must fit"

Multiplication of expressions exists because the world rarely hands us one term at a time. Area, volume, cost, and force are almost always products of quantities that each carry several parts.

Area and volume turn side lengths that are expressions into a single polynomial for the whole shape.
Expanding before solving is the step that turns a factored equation back into standard form so it can be analysed.
Physics and engineering models multiply quantities that are themselves sums, which is why a forgotten cross term changes the prediction.

What Are the Most Common Mistakes With Multiplying Expressions?

Mistake 1: Forgetting to distribute to every term

Where it slips in: Monomial-times-polynomial products, where the monomial reaches only the first term.

Don't do this: Writing 5a^2b^2(3a^2−4ab+6b^2) = 15a^4b^2 and stopping.

The correct way: Multiply the monomial by each term.

Mistake 2: Losing the negative sign

Where it slips in: Whenever a term inside a bracket is negative, or a whole expression is subtracted.

Don't do this: Treating −(3x+5) as −3x+5.

The correct way: Distribute the negative to every term: −(3x+5) = −3x−5.

Mistake 3: Adding exponents that shouldn't be added

Where it slips in: Multiplying powers of different bases, or confusing multiplication with addition of like terms.

Don't do this: Writing x^2 ⋅ y^3 = (xy)^5, or 3x + 4x = 12x.

The correct way: Add exponents only for the same base; leave different bases as a product; 3x + 4x = 7x.

Conclusion

Multiplying algebraic expressions means every term meets every term via the distributive property.
Apply the sign rule (like signs positive, unlike signs negative) and the exponent rule (a^m ⋅ a^n = a^{m+n}).
Use FOIL for two binomials and the standard identities for squares and differences of squares.
For polynomial products, multiply term by term, then combine like terms.
The most costly mistakes are dropped terms, lost negatives, and mis-added exponents.