Expand and Factor Linear Expressions: Paired Practice Guide

Algebra and Function Practice

Expanding and factoring linear expressions are inverse operations. Expanding multiplies a factor into a sum or difference; factoring pulls out the greatest common factor. This guide pairs both directions so you can check your work by reversing it. All numerical examples are hypothetical and chosen for easy arithmetic. Measure any real equipment yourself if you adapt the convention below.

Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.

The inverse relationship

A linear expression such as 3(2x + 5) is in factored form. Expanding gives 6x + 15. Factoring 6x + 15 by the greatest common factor returns 3(2x + 5). Because the two forms are equal for every value of x, you can verify one direction by performing the other.

The distributive property is the engine: a(b + c) = ab + ac. Factoring reverses it: ab + ac = a(b + c). For subtraction, a(b - c) = ab - ac. Keep signs attached to terms when you factor, especially when the GCF is negative.

1. Identify the factor outside parentheses or the GCF of all terms.
2. Multiply or divide each term carefully, keeping signs.
3. Simplify and write terms in a consistent order.
4. Reverse the operation to check equality.

Expanding step by step

To expand 4(3x - 2), multiply 4 by 3x to get 12x, then multiply 4 by -2 to get -8. The expanded form is 12x - 8. If the outside factor is negative, distribute the negative sign to every term: -2(5x + 3) = -10x - 6.

Watch for a common factor that is a fraction or a variable. For example, (1/2)(6x + 4) = 3x + 2. Expanding is complete when no parentheses remain and like terms are combined.

1. Write the outside factor times each inside term.
2. Compute each product, noting sign rules.
3. Remove parentheses and combine like terms if any.
4. Check by factoring the result mentally.

Factoring by greatest common factor

To factor 18x + 12, list factors of 18 and 12. The greatest common factor is 6. Divide each term by 6: 18x ÷ 6 = 3x and 12 ÷ 6 = 2. The factored form is 6(3x + 2). Always include the sign: -15x + 10 has GCF -5 if you prefer a positive leading term inside, giving -5(3x - 2). Either -5(3x - 2) or 5(-3x + 2) is valid; choose one convention and state it.

When a variable is common, include it. For 8x² + 4x, the GCF is 4x, so the factored form is 4x(2x + 1). Check by expanding: 4x(2x + 1) = 8x² + 4x.

1. Find the GCF of all numerical coefficients and any common variable power.
2. Divide each term by the GCF, keeping signs.
3. Write the GCF outside parentheses and the quotients inside.
4. Expand to verify you recover the original expression.

Convention for handwritten work

For handwritten algebra, use a consistent convention: write the variable in lower case, keep the multiplication dot optional, and align equals signs vertically. X-height is measured from baseline to midline; letter styles vary, so accept any legible lower-case x that is not confused with a multiplication sign. In this guide, a handwritten x is valid if its height is between the baseline and the midline of the writing line, and if it is distinguishable from a plus sign. This convention allows valid alternatives rather than a single universal answer.

When checking a partner's work, apply the same convention: confirm the expression is legible, the distributive step is shown, and the final form is equivalent. Do not penalize a different but equivalent factored form, such as 5(-3x + 2) instead of -5(3x - 2), if the expansion matches.

1. Write each expression on one line with clear spacing.
2. Show the distributive or division step.
3. Mark the final answer and expand or factor once to check.
4. Use the x-height convention only for legibility, not for grading equivalence.

Worked example

Expand -4(3x - 5) and then factor the result by the GCF to check.

Four practice exercises

Common mistakes

• Distributing only to the first term: 3(2x + 5) written as 6x + 5. Multiply the outside factor by every term inside: 3·2x = 6x and 3·5 = 15, so 6x + 15.
• Losing a negative sign when factoring: -6x - 9 written as 3(2x + 3). Choose the GCF with its sign or adjust inside: -3(2x + 3) expands to -6x - 9, so use -3(2x + 3).
• Factoring only the numerical GCF and ignoring a common variable: 8x² + 4x written as 4(2x² + x). Include the common variable: GCF is 4x, so 4x(2x + 1).
• Assuming a factored form is unique and marking an equivalent form wrong. Accept any form that expands to the original, such as 5(-3x + 2) or -5(3x - 2).

Self-check rubric

• Expansion multiplies the outside factor by every term inside.
• Factoring divides every term by the stated GCF.
• Signs are preserved and checked by expansion.
• Equivalent factored forms are accepted if expansion matches.
• Handwriting follows the stated x-height convention for legibility.

Questions and answers