Foil & Reverse-Foil, Algebra for GMAT - 30 Days Preparation for GMAT PDF

FOIL & REVERSE-FOIL

FOIL is an acronym standing for First, Outer, Inner, Last. It is a method for multiplying two binomials and is especially useful when working with quadratics. Many standardised-test questions require quick expansion (FOIL) or factoring (reverse FOIL) of quadratic expressions. Be alert for squared terms or expressions under a square root-these often indicate a quadratic structure.

The FOIL method explained

To multiply two binomials, multiply each pair of terms indicated by the words First, Outer, Inner, and Last, then collect like terms.

• First: product of the first terms of each binomial.
• Outer: product of the outer terms.
• Inner: product of the inner terms.
• Last: product of the last terms.

FOIL example

Example: Expand (x + 3)(x + 3).

(x + 3)(x + 3)
x × x = x²
x × 3 = 3x
3 × x = 3x
3 × 3 = 9

Collect like terms:

x² + 3x + 3x + 9
x² + 6x + 9

This can be recognised as a perfect square: (x + 3)² = x² + 6x + 9.

Reverse FOIL (factoring)

Reverse FOIL means converting a quadratic expression into a product of two binomials. The common approach for quadratics of the form x² + bx + c is to find two numbers whose product is c and whose sum is b.

Reverse FOIL example

Example: Factor x² + 6x + 9.

x² + 6x + 9

Find two numbers that multiply to 9 and add to 6: these are 3 and 3.

Rewrite the quadratic as (x + 3)(x + 3) or (x + 3)².

When the coefficient of x² is not 1, use factoring by grouping or apply the quadratic formula when necessary.

The three classic quadratics

These common forms recur frequently; it is efficient to memorise their expanded forms and factored forms.

• (x + y)(x + y) - perfect square.
• (x - y)(x - y) - perfect square with negative cross term.
• (x + y)(x - y) - difference of squares.

Classic quadratic expansions (worked)

1. Expand (x + y)(x + y).
(x + y)(x + y)
x × x = x²
x × y = xy
y × x = yx = xy
y × y = y²
Collect like terms:
x² + xy + xy + y²
x² + 2xy + y²
2. Expand (x - y)(x - y).
(x - y)(x - y)
x × x = x²
x × (-y) = -xy
(-y) × x = -yx = -xy
(-y) × (-y) = y²
Collect like terms:
x² - xy - xy + y²
x² - 2xy + y²
3. Expand (x + y)(x - y).
(x + y)(x - y)
x × x = x²
x × (-y) = -xy
y × x = yx = xy
y × (-y) = -y²
Collect like terms:
x² - xy + xy - y²
x² - y²

Symbolism

 (custom symbols / operators)

A symbolism question defines a new operator and requests calculations using that definition. Read the operator definition carefully and apply it exactly as stated. These questions test accuracy and attention to detail.

Symbolism example (worked)

Given: aΩb = ab + a²

Compute: 2Ω3

Replace a with 2 and b with 3 in the definition:

2Ω3 = 2 × 3 + 2²

2 × 3 = 6

2² = 4

6 + 4 = 10

Practical advice and application

When you see a binomial product or a quadratic expression, decide quickly whether to expand (FOIL) or to factor (reverse FOIL). Look for:

• squared binomials that match (x ± y)² patterns;
• constants whose factor pairs sum to the coefficient of x;
• structure of a difference of squares x² - y² that factors as (x + y)(x - y).

Practice by expanding and factoring a variety of examples, including cases where the coefficient of x² is not 1, and by solving short timed drills that mix FOIL, reverse FOIL and symbolic-operator problems. This builds both speed and accuracy for test situations.