Algebra Practice

Expanding Expressions Practice Test

Use distribution, binomial multiplication, and algebraic identities to remove parentheses and combine like terms.

Expanding Expressions Practice Test

20 expansion questions from single distribution through cubic identities.

Instant feedback · Worked explanations
After the test · expansion workshop

Expanding an expression removes grouping without changing the value

Brackets hide multiplication. Expanding makes those products explicit, which often reveals like terms that can be combined afterward. The important habit is completeness: every factor in one group must interact with the correct term or terms in the other group, and every sign must survive that multiplication.

Single bracketsNegative factorsDouble brackets Nested groupingArea modelsFractionsDecimalsReverse checks
Single-bracket fan

One outside factor must multiply every term inside the bracket

× 3
Inside term2x
Inside term−5
Expanded result6x − 15
Negative-factor expansion

A negative multiplier changes signs through ordinary multiplication

−2(3x − 7)
−6x + 14
(−2)(3x) = −6x
(−2)(−7) = +14

A minus sign outside a bracket can be treated as a factor of −1 or as part of a larger negative coefficient. Either way, it must multiply every term inside.

If only one sign changes after expanding a two-term bracket with an outside negative factor, recheck the distribution.
Area-tile model

Double-bracket expansion can be seen as the area of a partitioned rectangle

For (x + 3)(x + 2), every row part multiplies every column part.

x · x = x²
x · 2 = 2x
3 · x = 3x
3 · 2 = 6
Total: x² + 5x + 6
Double-bracket expansion

Track all four products before combining the middle terms

Product 1x · x = x²
Product 2x · (−3) = −3x
Product 34 · x = 4x
Product 44 · (−3) = −12

Then combine the like middle terms: x² − 3x + 4x − 12 = x² + x − 12.

Nested-bracket sequence

When grouping is nested, simplify the inner structure before expanding the outer factor

Start2[3x − (x − 4)]
Inner bracket2[3x − x + 4]
Collect inside2[2x + 4]
Expand outside4x + 8
Term-count checkpoint

Counting expected products can reveal a missing term before you finish

2 terms × 2 terms

(a + b)(c + d)
→ ac + ad + bc + bd
Expect four raw products before any like terms combine.

1 factor × 3 terms

k(a + b + c)
→ ka + kb + kc
Expect three products. If only two appear, one inside term was skipped.
Fractions and decimals

Non-integer factors change the arithmetic, not the expansion rule

Fractional factor

3/4(8x − 12)
= 6x − 9

Decimal factor

0.6(5y + 10)
= 3y + 6
4(x + 3)
↓ expand
4x + 12
↑ factor
4(x + 3)
Reverse-check by factoring

A quick factor check can confirm that expansion preserved the structure

If the expanded expression has an obvious common factor, pulling it back out should recover the original bracketed form.

Expand: 4(x + 3) → 4x + 12
Factor back: 4x + 12 → 4(x + 3)
Matching the start is a strong structural check.
Expansion error audit

Four shortcuts that produce incomplete or non-equivalent expansions

3(x + 4) = 3x + 4 3(x + 4) = 3x + 12

The outside factor must multiply both terms.

(x + 2)² = x² + 4 (x + 2)² = x² + 4x + 4

Squaring a binomial means multiplying two copies of the entire binomial.

(x + 3)(x − 2) = x² − 6 (x + 3)(x − 2) = x² + x − 6

The two middle products −2x and +3x must both be included.

−2(x − 5) = −2x − 10 −2(x − 5) = −2x + 10

The product of two negative factors is positive.

Use the score diagnostically

A missed expansion question usually points to one of four structural weaknesses

Classify the mistake before repeating the test so the next attempt targets the correct step.

CoverageOne or more required products were skipped.
SignsA negative product changed incorrectly.
Nested structureInner grouping was not handled before the outer factor.
CollectionThe expansion was correct, but like terms were combined incorrectly.