How to multiply binomials using the FOIL method

Multiplying binomials is a core algebra skill that appears in classroom exercises, tests and problems involving quadratic expressions. A binomial is an algebraic expression with two terms, such as (x+3), (2x-5) or (a-b). When two binomials are multiplied, each term in the first bracket must be multiplied by each term in the second.

The FOIL method provides a reliable order for completing this task. It means First, Outer, Inner, Last, helping students organise four multiplication steps before collecting like terms. The method is especially useful when expanding expressions that will later be rewritten as quadratic equations or vertex-form problems.

What each FOIL letter represents

Consider the expression:

[ (x+4)(x+7) ]

The first terms are (x) and (x), so the First multiplication is:

[ x \times x=x^2 ]

The Outer terms are (x) from the first bracket and (7) from the second:

[ x \times 7=7x ]

The Inner terms are (4) and (x):

[ 4 \times x=4x ]

Finally, multiply the Last terms, (4) and (7):

[ 4 \times 7=28 ]

Write all four results in order:

[ x^2+7x+4x+28 ]

The expression is not fully simplified until the like terms (7x) and (4x) are combined. This gives:

[ x^2+11x+28 ]

FOIL is simply a memory aid for the distributive property. It does not change the rules of multiplication, and it should only be used when each bracket contains exactly two terms.

A step-by-step example with signs

Negative signs require careful attention. Expand:

[ (2x-3)(x+5) ]

First, multiply the first terms:

[ 2x \times x=2x^2 ]

Next, multiply the outer terms:

[ 2x \times 5=10x ]

Then multiply the inner terms:

[ -3 \times x=-3x ]

The negative sign belongs to the number 3, so the inner product is negative. For the last terms:

[ -3 \times 5=-15 ]

Put the four products together:

[ 2x^2+10x-3x-15 ]

Combine the like terms:

[ 2x^2+7x-15 ]

The final expression is (2x^2+7x-15). A useful check is to substitute a simple value, such as (x=1), into both the original and expanded expressions. The original gives ((2-3)(1+5)=-6), while the expanded version gives (2+7-15=-6), confirming that the expansion is correct.

Keeping unlike terms separate

Students sometimes add terms that look similar but are not like terms. For example, (x^2), (x) and a constant have different powers or no variable at all. They cannot be combined.

In the expression:

[ 3x^2+8x+2x^2-5x+6 ]

combine the squared terms with squared terms and the single-(x) terms with single-(x) terms:

[ (3x^2+2x^2)+(8x-5x)+6 ]

This becomes:

[ 5x^2+3x+6 ]

The same principle applies after using FOIL. In an Australian Year 9 or Year 10 classroom, a teacher may award separate marks for expansion and simplification, so showing each product can make the working easier to follow. It also helps identify whether an error came from multiplication, signs or collecting terms.

Checking algebra in practical settings

Algebra can describe practical calculations involving measurements, prices and changing quantities. Suppose a rectangular garden in Melbourne has a length of (x+3) metres and a width of (x+2) metres. Its area is:

[ (x+3)(x+2) ]

Using FOIL:

[ x^2+2x+3x+6=x^2+5x+6 ]

The result is in square metres because area is length multiplied by width. This connection makes it easier to see why the squared term appears: (x) metres multiplied by (x) metres produces (x^2) square metres.

A similar example could involve a Sydney market stall selling packs priced at (x+4) dollars, with a quantity represented by (x+2). The algebraic product is still expanded in the same way, although real prices would need to be written with Australian dollars and cents. If a calculation includes GST, Australia’s 10 per cent goods and services tax must be applied according to the relevant tax rules rather than treated as an extra algebra term by accident.

When FOIL is not the best method

FOIL works for two binomials, but it is not the most suitable method for every expression. For a binomial multiplied by a trinomial, such as:

[ (x+2)(x^2+3x+4) ]

use the distributive property:

[ x(x^2+3x+4)+2(x^2+3x+4) ]

Then expand each part:

[ x^3+3x^2+4x+2x^2+6x+8 ]

Collect like terms:

[ x^3+5x^2+10x+8 ]

The area model is another useful approach. Draw a grid, place the terms of one bracket across the top and the terms of the other down the side, then fill each cell with a product. This method is helpful for visual learners and reduces the chance of omitting a term.

FOIL also works in reverse when factoring some quadratic expressions. For example:

[ x^2+9x+20 ]

can be written as:

[ (x+4)(x+5) ]

because the outside and inside products combine to (9x), while the last terms multiply to (20). Factoring is the reverse process of expansion, so checking the answer with FOIL confirms whether the factorisation is valid.

Use a structured four-line layout when practising:

[ \text{First},\quad \text{Outer},\quad \text{Inner},\quad \text{Last} ]

Write the products, combine like terms, and then check the result by substitution. Whether the work is completed in a Brisbane classroom, during homework after school, or with an online algebra calculator, the next concrete step is to expand ((3x+2)(x-6)) and simplify every term.