Choose and apply the correct factoring method for a wide range of polynomial expressions.
20 mixed factoring questions from basic GCFs through cubic patterns.
Factoring Polynomial Algebra · Test 1 of 10
This mixed practice test moves from basic greatest common factors through trinomials, special products, grouping, cube identities, repeated patterns, cubic expressions, and factored equations. The main challenge is not memorizing one procedure — it is deciding which factoring method fits the expression and continuing until every factor is completely reduced.
Factoring method selector
Look for a numerical, variable, or combined factor shared by every term. Factor it first.
Two terms may suggest special products. Three terms suggest a trinomial. Four terms often suggest grouping.
One successful factoring step may reveal another difference of squares, GCF, trinomial, or special product.
Method atlas
6x³ + 9x² = 3x²(2x + 3)
Always check this first, even when another pattern is also present.
x² + bx + c → (x + m)(x + n)
For monic trinomials, m+n=b and mn=c.
a² − b²
a² ± 2ab + b²
Recognize the pattern before using a slower general method.
ax + ay + bx + by
Create a common binomial factor by grouping compatible terms.
a³ ± b³
Cube identities use different sign patterns from square identities.
ax² + bx + c, a ≠ 1
Factor pairs must reproduce both the leading term and the middle coefficient.
u² − 9 → (u − 3)(u + 3)
A complicated repeated expression can be treated as one temporary unit.
AB = 0 → A = 0 or B = 0
After factoring an equation, use the zero-product property to solve it.
Trinomial product-and-sum logic
Need two numbers whose product is 12 and whose sum is 7.
A pair with the correct product but wrong sum is not a valid factorization.
Special-product recognition
a² − b² = (a − b)(a + b)
Requires subtraction between two perfect squares.
a² + 2ab + b² = (a + b)²
The middle term is twice the product of the square roots of the outer terms.
a² − 2ab + b² = (a − b)²
The sign inside the binomial follows the sign of the middle term.
Cube identities · sign checkpoint
a³ + b³ = (a + b)(a² − ab + b²)
The binomial keeps the plus sign; the middle term in the quadratic factor is negative.
a³ − b³ = (a − b)(a² + ab + b²)
The binomial keeps the minus sign; the middle term in the quadratic factor is positive.
Factoring by grouping
Pair terms that can produce the same inner factor.
Each group now contains x+y.
Factor out the repeated binomial.
Grouping may expose another recognizable pattern.
Factor completely
The page explicitly warns against stopping too early. After every factoring step, inspect every factor again.
2(x² − 9)
The GCF is removed, but the difference of squares remains.
x² − 9 = (x − 3)(x + 3)
The inside factor is still reducible.
2(x − 3)(x + 3)
Now each factor has been checked again.
From factoring to solving
The product is zero, so at least one factor must be zero.
Factoring becomes a solving method only after each factor equation is handled.
Common mistakes from the page
This often simplifies the remaining expression and reveals the next method.
A factor may still contain a GCF, special product, trinomial, or repeated pattern.
The final term b² remains positive in both cube identities.
The middle term is the fastest check against a wrong factor pair.
Final factoring checklist