Algebra Practice

Distributive Property Practice Test

Expand, factor, simplify, and solve expressions using distribution in varied forms.

Distributive Property Practice Test

20 distributive-property questions with instant feedback.

Instant feedback · Worked explanations
After the test · distribution operator

The distributive property sends one multiplier to every term inside a group

Distribution connects multiplication with addition and subtraction. It can expand grouped expressions, simplify equations, expose like terms, and run in reverse when you factor out a common factor. The key is structural: the outside factor acts on the entire grouped sum or difference, not only the first visible term.

Positive factorsNegative factorsFractionsDecimals Reverse distributionEquationsMultiple variablesApplications
Multiplier fan

Visualize distribution as one source branching to each term

For 4(x + 3), the multiplier 4 creates two separate products before the results are added.

× 4
Branch 14 · x = 4x
Branch 24 · 3 = 12

Result: 4(x + 3) = 4x + 12.

Sign propagation

Negative distribution is multiplication by a negative number

−3(2x − 5)
−6x + 15
because (−3)(−5) = +15

The negative sign is not a special decoration. It belongs to the outside factor and therefore changes the sign of each product according to ordinary multiplication rules.

A common failure is to distribute the magnitude correctly but leave the second sign unchanged.
Area model

Distribution is built into the geometry of a split rectangle

A rectangle of height a and total width b + c can be split into two smaller rectangles.

Total width: b + c
Total area: a(b + c)
Split areas: ab and ac
Therefore: a(b + c) = ab + ac
Reverse distribution

Factoring a common factor is distribution run backward

Instead of multiplying one factor into several terms, identify what every term already shares and pull it outside.

Expanded 12x + 18
↔
Factored 6(2x + 3)
Fractions and decimals

The structure stays the same even when the outside factor is not an integer

Fractional factor

½(8x − 6)
= 4x − 3

Decimal factor

0.4(5y + 15)
= 2y + 6

Only the coefficient arithmetic changes; every term still receives the outside multiplier.

Distribution inside equations

After expansion, an equation can reveal one solution, no solution, or infinitely many solutions

One solution 2(x + 3) = 14
2x + 6 = 14 → x = 4
No solution 3(x + 2) = 3x + 10
3x + 6 = 3x + 10 → 6 = 10
Infinitely many 4(x − 1) = 4x − 4
4x − 4 = 4x − 4 → identity
Distribution plus collection

Distribution often creates like terms that were hidden by parentheses

Original3(2x − 5) + 4x
Distribute6x − 15 + 4x
Combine10x − 15
Checksame value for every x
Event cost model 3 groups
each costs
(12 + 4n)
Distribution in a cost model

A grouped real-world cost can be expanded into fixed and variable parts

If each of 3 groups costs 12 dollars plus 4 dollars per person, the total expression is 3(12 + 4n).

3(12 + 4n)
= 36 + 12n
$36 is the combined fixed part
$12n is the combined per-person part
Distribution audit

Four shortcuts that break equivalence

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

The multiplier must reach both terms inside the parentheses.

−(x − 4) = −x − 4 −(x − 4) = −x + 4

Multiplying by −1 changes both signs.

3(x + y) = 3x + y 3(x + y) = 3x + 3y

Every term inside the group gets the factor 3.

6x + 12 = 3(2x + 12) 6x + 12 = 3(2x + 4)

When factoring 3 out, each inside term must be divided by 3.

Use the score diagnostically

A missed distribution question usually belongs to one of four different structural errors

Classify the error before repeating the test so the next attempt targets the exact weak point.

CoverageThe factor did not multiply every inside term.
SignsA negative product or subtraction changed incorrectly.
Reverse distributionThe common factor or inside quotient was wrong.
After expansionLike terms or equation classification was handled incorrectly.