Algebra Practice

Factoring Common Factors Practice Test

Recognize repeated binomial or polynomial factors and factor them using the distributive property.

Factoring Common Factors Practice Test

20 questions on common binomials, polynomial factors, and complete factoring.

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Factoring Polynomial Algebra · Common Factors

The greatest common factor is built from the largest shared coefficient and the lowest shared powers

This practice set focuses on extracting common factors accurately and completely. The questions use numerical GCFs, variable GCFs, several variables at once, negative leading terms, decimal and fractional coefficients, exact-GCF identification, and checks for whether a proposed factorization really uses the greatest common factor. The page also emphasizes that a common factor can be more than a single monomial: it may be a repeated binomial, squared expression, or larger polynomial.

numerical GCFlowest powersmultiple variables negative GCFdecimalsfractions complete factorizationrepeated expression factors
A valid common factor is not automatically the greatest common factor. The final inside polynomial should not still share another factor across every term.

GCF extraction bench

Build the outside factor in separate layers

Coefficient layer

18 and 24 → GCF = 6

Find the greatest number dividing every coefficient.

Variable layer

x⁴ and x³ → shared x³

For each variable present in every term, choose the lowest exponent.

Combine the layers

12 · x³ · y² → 12x³y²

The complete GCF contains every numerical and variable part shared by all terms.

Quick GCF shelf

Compact rules for common-factor problems

Coefficient GCF Use the greatest common divisor GCF(18, 24) = 6

Do not stop at a smaller valid common divisor such as 2 or 3.

Variable GCF Use the minimum shared exponent GCF(x⁴, x³) = x³

A higher exponent would not divide every term.

Several variables Apply the minimum-exponent rule separately GCF(x⁴y², x³y⁵) = x³y²

Each variable is scanned independently.

Distributive check Verify by multiplying back g(A + B) = gA + gB

Distribution should recreate every original term exactly.

Negative GCF Useful when the leading terms are negative −14x² + 21x = −7x(2x − 3)

Factoring out a negative can make the inside expression easier to read.

Common expression The factor need not be one term P(x)A + P(x)B = P(x)(A + B)

A repeated binomial or larger polynomial can play the same structural role as a monomial GCF.

Expression
30m⁴n³ + 45m²n⁵ − 75m³n²
Coefficients
GCF(30, 45, 75) = 15
m powers
4, 2, 3 → choose m²
n powers
3, 5, 2 → choose n²
Full GCF
15m²n²

Lowest-power rule

Variable exponents are limited by the term with the smallest power

Compare x

x⁶ and x⁴
lowest power = x⁴

x⁵ cannot be common because it does not divide the x⁴ term.

Compare y

y, y², y³
lowest power = y

A single y is the largest power shared by all three terms.

Missing variable

12x² + 8x − 4
x is not common

The constant term contains x⁰, so no positive power of x belongs in the GCF.

Several variables at once

Treat coefficient, x, y, and any other variable as separate GCF channels

1 · Expression 32x³y − 48x²y²

Both terms share numerical and variable factors.

2 · Coefficient GCF(32,48) = 16

Use the greatest common numerical factor.

3 · Variables x² and y

Minimum exponents are x² and y¹.

4 · Factor 16x²y(2x − 3y)

Distribute back to verify both original terms.

Negative-GCF choice

Factoring out a negative can produce a cleaner inside polynomial

Original expression

−36x³ − 24x² − 12x

Every term is negative and every term contains 12x.

FACTOR
−12x

Cleaner factorization

−12x(3x² + 2x + 1)

The terms inside the parentheses become positive.

Decimals and fractions

The same GCF idea applies when the coefficients are not whole numbers

Decimal coefficients

0.6x² + 1.8x

= 0.6x(x + 3)

The common decimal coefficient is 0.6, and both terms contain x.

Fractional coefficients

(3/4)x² + (9/8)x

= (3/8)x(2x + 3)

After choosing the fractional GCF, divide each coefficient exactly.

Common factor versus greatest common factor

A factorization can be equivalent and still fail the GCF instruction

This distinction is tested directly: the outside factor must be the greatest common factor, not merely any shared factor.

Valid but incomplete choice 16x² + 20x = 2x(8x + 10)

Distribution works, but 2x is not the greatest common factor.

Find the actual GCF GCF(16,20) = 4
shared variable = x

The full GCF is 4x.

Preferred factorization 16x² + 20x = 4x(4x + 5)

The terms inside no longer share another common factor.

After the GCF comes out

A GCF can expose another factoring pattern

Start

64x⁶ − 16x⁴

The full common factor is 16x⁴.

GCF removed

16x⁴(4x² − 1)

The inside factor is now visibly a difference of squares.

If asked to factor completely

16x⁴(2x − 1)(2x + 1)

Always follow the wording: “factor out the GCF” and “factor completely” are not identical instructions.

A common factor can be an expression

The distributive pattern works with repeated binomials or larger polynomial factors too

Repeated factor visible

(x + 2)A + (x + 2)B

The shared object is the whole binomial x+2.

DISTRIBUTIVE
PROPERTY

Factor the repeated expression

(x + 2)(A + B)

The same logic works when the repeated factor is squared or is a larger polynomial.

6x
3x + 5
18x² + 30x
square units

Geometric interpretation from the test

Factoring an area polynomial can reveal possible side expressions

If the area is 18x² + 30x and one side is the greatest common factor, factor the area expression. The GCF 6x becomes one side, and the remaining factor 3x + 5 becomes the other.

18x² + 30x = 6x(3x + 5)

Diagnostic mistakes

Most common-factor errors are selection errors, not distribution errors

Smaller common factor chosen
Using 2x when 4x divides every term. Check whether the inside terms still share another factor.

If they do, the outside factor was not greatest.

Highest exponent used
Using x⁴ as the GCF of x⁴ and x³. Use x³, the lowest shared power.

The GCF must divide every term without leaving negative exponents.

Variable included despite constant
Putting x in the GCF of 12x² + 8x − 4. The constant term contains no x, so x is not common to all terms.

Every term must participate in the common factor.

Negative factor mishandled
Factoring out −7x without changing the inside signs correctly. Divide each original term by the exact negative GCF.

A quick distribution check catches sign mistakes immediately.

Stopped before complete factoring
Assuming that removing the GCF always finishes the problem. If the instruction says “factor completely,” inspect the remaining factor again.

A difference of squares or another pattern may appear only after the GCF is removed.

Final common-factor checklist

Before choosing an answer, audit the outside factor and every inside term

Greatest coefficient used The numerical part is the greatest common divisor of all coefficients, not merely a convenient divisor.
Lowest powers used Each shared variable appears with the smallest exponent found across all terms.
Every term checked A constant or lower-power term has not been ignored when building the common factor.
Distribution + completeness checked Multiplying back restores the original polynomial, and further factoring is done when the instruction requires it.