What does it mean to expand an algebraic expression?
Expansion rewrites a product or power without the original multiplication grouping. For example, 3(x + 4) expands to 3x + 12. The expanded and factored forms are equivalent; they simply reveal different features of the same polynomial.
Expanded standard form is useful for combining sums, identifying coefficients, and comparing polynomials. Factored form is often better for seeing zeros. Expanding does not solve for x unless the expression is part of an equation and a solving method follows.
How to expand an expression step by step
Treat subtraction as a signed coefficient and keep grouped factors visible until every required product has been written. Only then collect like powers.
- Identify each factor and the terms contained in it.
- Distribute a monomial to every term inside its parentheses.
- For two polynomials, multiply every term in the first by every term in the second.
- Multiply coefficients and apply exponent rules to matching variable bases.
- Write all partial products with their correct signs.
- Combine like terms and arrange the final polynomial in descending-power order.
- Check equivalence by factoring back or evaluating both forms at sample values.
Worked example: (x + 3)(x − 2)
Multiply x by both terms in the second factor: x(x − 2) = x² − 2x. Then multiply 3 by both terms: 3(x − 2) = 3x − 6. Add the partial products to get x² − 2x + 3x − 6.
Combine the linear terms −2x + 3x = x. The expanded result is x² + x − 6. A check at x = 4 gives (7)(2) = 14 in the original form and 16 + 4 − 6 = 14 in the expanded form.
FOIL is a shortcut, not a separate algebra rule
FOIL stands for First, Outer, Inner, Last and helps organize the four products of two binomials. It is simply the distributive property used twice. Because it is designed for two-term factors, it should not be used as-is for a binomial multiplied by a trinomial.
The broader rule always works: multiply every term in the first polynomial by every term in the second. A binomial times a trinomial begins with six partial products before like terms are combined.
Expanding powers and handling negative signs
A power such as (x + 2)² means (x + 2)(x + 2), not x² + 4. The middle term 4x comes from the two cross-products. Likewise, (x − 2)² = x² − 4x + 4; the square does not make the middle sign positive.
When a negative monomial is outside parentheses, distribute its sign to every term. For −2x(x² − 3x + 1), the result is −2x³ + 6x² − 2x.
Common expansion mistakes
Distributing to only the first term, omitting a cross-product, and combining unlike powers are the main errors. Another is adding exponents during addition rather than during multiplication of equal bases.
Count expected partial products before combining. Then inspect the leading term, constant term, and middle coefficients. The calculator’s equivalent-expression check confirms the collected result against the original supported product.