Greatest Common Factor Factoring Practice Test

Factoring Polynomial Algebra · GCF Factoring

Greatest common factor factoring is an audit: coefficient gcd, minimum exponents, exact division, then completeness

This practice set focuses specifically on greatest common monomial factors. Questions include numerical gcds, shared variable powers, negative leading terms, three-term polynomials, accurate term-by-term division, and recognizing when a proposed GCF factorization is still incomplete. The method is also the first factoring check to make before attempting trinomials, special products, grouping, or other polynomial techniques.

coefficient gcdsmallest exponentsnegative GCF three-term polynomialsdivide every termcomplete factorization

The GCF is the largest monomial that divides every term. If the terms inside parentheses still share another factor, the extraction was not complete.

GCF audit ladder

Use the same four checks every time

Coefficient Find the greatest numerical divisor of all coefficients.

This is the gcd part of the GCF.

Variables For each shared variable, choose the smallest exponent.

A higher power will fail to divide at least one term.

Divide Divide every term by the full GCF.

Do not skip signs, coefficients, or variable powers.

Audit Check the inside polynomial for another common factor.

If one remains, the factorization is incomplete.

Quick GCF shelf

Compact rules for numerical and variable greatest common factors

Numerical gcd Use the greatest divisor shared by all coefficients gcd(18, 30, 42) = 6

A smaller shared divisor is valid but not greatest.

Variable rule Use the lowest exponent appearing in every term GCF(x⁵, x³, x²) = x²

The smallest power limits the shared monomial.

Full GCF Multiply numerical and variable parts gcd part × shared variables

For several variables, apply the minimum-exponent rule separately to each one.

Distribution check Multiply back to verify g(A + B + C) = gA + gB + gC

The product must reproduce every original term exactly.

Coefficient gcd

The numerical part must be greatest, not merely common

Common divisor

24 and 36
6 divides both

So 6 is a common divisor.

But check larger divisors

12 divides 24 and 36

Therefore 6 is not the greatest common divisor.

Numerical GCF

gcd(24,36) = 12

This is the coefficient part that belongs in the full GCF.

Smallest shared exponents

The lowest variable power is the largest power guaranteed to divide every term

Single variable

x⁵, x³, x²
→ x²

x³ is too large because it does not divide the x² term.

Two variables

x⁴y³, x²y⁵, x³y²
→ x²y²

Scan x and y independently.

Missing variable

12x² + 18x + 6
→ no x in the GCF

The constant term contains x⁰, so no positive x-power divides every term.

Full extraction walkthrough

START

42x⁵ − 28x³ + 14x² Three terms must all be included in the GCF scan.

NUMERIC

gcd(42,28,14) = 14 14 is the greatest common numerical divisor.

VARIABLE

x⁵, x³, x² → x² The smallest shared exponent is 2.

FACTOR

14x²(3x³ − 2x + 1) Every original term has been divided by the full GCF.

Negative leading terms

A negative GCF can make the factor inside easier to read

Original polynomial

−18x³ − 12x² − 6x

The full positive monomial GCF is 6x, but a negative choice may be cleaner.

FACTOR
−6x

Cleaner inside polynomial

−6x(3x² + 2x + 1)

Every inside coefficient is positive, reducing sign clutter.

Three-term GCF discipline

Every term participates in each GCF decision

1 · Coefficients 30, 45, 75

The greatest shared divisor is 15.

2 · First variable m⁴, m², m³ → m²

The smallest exponent is 2.

3 · Second variable n³, n⁵, n² → n²

Again use the minimum exponent.

4 · Combine 15m²n²

This is the greatest common monomial.

Incomplete factorization detector

Equivalent does not always mean fully factored by the GCF

One of the skills on the page is recognizing whether a GCF factorization is complete.

Incomplete 16x² + 20x = 2x(8x + 10)

The factorization distributes correctly, but the inside coefficients still share 2.

Audit inside 8x and 10 still share 2

This proves the outside factor was not greatest.

Complete GCF extraction 16x² + 20x = 4x(4x + 5)

The terms inside parentheses now share no common monomial factor greater than 1.

Divide every term accurately

After choosing the GCF, each inside term is an exact quotient

Original term

30x⁴ ÷ 6x² = 5x²

Divide coefficients and subtract variable exponents.

Second term

−18x³ ÷ 6x² = −3x

Preserve the negative sign.

Third term

12x² ÷ 6x² = 2

A term can reduce to a constant inside the parentheses.

Final completeness check

The inside polynomial should fail a second GCF scan

Check coefficients

gcd(5,3,2) = 1

No numerical factor greater than 1 remains common.

Check variables

5x² − 3x + 2

The constant term means x is not shared by every term.

Conclusion

GCF extraction is complete

No additional common monomial can be factored from the entire inside expression.

Common mistakes from the page

The most common GCF errors happen during selection, sign handling, and the final audit

Common but not greatest

Factoring out 2x when every term is divisible by 4x. Use the greatest shared coefficient and all shared variable factors.

The inside polynomial is a useful diagnostic: if another GCF remains, the first extraction was incomplete.

Largest exponent used

Choosing x⁵ from terms containing x⁵, x³, and x². Use x², the smallest shared exponent.

The GCF must divide every term.

Sign lost

A negative term becomes positive after division without a negative factor accounting for it. Divide each signed term by the exact signed GCF.

Distributing back is the quickest sign check.

Another common factor left inside

Stopping at 2x(8x + 10). Recognize that 8x and 10 still share 2.

For GCF factoring, the inside expression should not retain another factor common to every term.

Final GCF factoring checklist

Before choosing an answer, verify all four parts of the monomial GCF

Greatest coefficient The numerical factor is the gcd of all coefficients, not merely a smaller common divisor.

Smallest exponents Every shared variable uses the lowest exponent appearing across all terms.

Exact division Every term is divided by the full GCF with signs and exponent subtraction handled correctly.

Inside polynomial audited No additional common monomial factor remains inside the parentheses.