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Factoring practice

Practice factoring by recognizing structure

Work through 10 fixed problems that begin with a greatest common factor and progress to trinomials, grouping, and repeated special-product patterns.

Focused practice

Work through 10 checked problems

Use greatest common factors, trinomial patterns, grouping, and repeated differences of squares.

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Interactive practice

Factoring practice: 10 checked problems

Use greatest common factors, trinomial patterns, grouping, and repeated differences of squares.

Question 1 of 101 of 10

Greatest common factor

Question 1

Factor completely over the integers.

12x3−18x212x^3-18x^2

Coverage

Skills in this practice collection

  1. Foundation

    Start with the GCF

    Extract the greatest shared coefficient and the lowest shared exponent before using a more specific pattern.

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  2. Core skill

    Match a product pattern

    Use product-and-sum reasoning for trinomials and identify perfect squares or differences of squares.

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  3. Challenge

    Factor in more than one stage

    Use grouping or an outer pattern, then inspect every resulting factor again.

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Sample problem

See the expected explanation depth

Factor completely and verify by expansion.

4x3−20x2+24x4x^3-20x^2+24x
  1. 1
    Extract the greatest common factor

    Every term contains 4x.

    4x(x2−5x+6)4x(x^2-5x+6)
  2. 2
    Factor the trinomial

    Negative 2 and negative 3 multiply to positive 6 and add to negative 5.

    4x(x−2)(x−3)4x(x-2)(x-3)
  3. 3
    Expand to check

    First multiply the binomials, then distribute 4x.

    4x(x2−5x+6)=4x3−20x2+24x\begin{aligned}4x(x^2-5x+6)\\&=4x^3-20x^2+24x\end{aligned}

Answer

4x(x−2)(x−3)4x(x-2)(x-3)

Study plan

Use mistakes to choose the next problem

The correct method is usually visible in the polynomial's structure. Use this order every time so a common factor or unfinished factor is not missed.

  1. Check the greatest common factor

    Compare coefficients and exponents across all terms before looking for a named pattern.

  2. Count terms and inspect signs

    Two terms may form a square or cube pattern; three terms may form a trinomial; four terms may support grouping.

  3. Factor each result again

    A first factorization can reveal another difference of squares or a remaining common factor.

  4. Expand every factor

    Match the degree, signs, and every coefficient with the original polynomial.

Common questions

Practice factoring by recognizing structure FAQ

What is the first step in every factoring problem?

Look for a greatest common factor across every term. Removing it first makes the remaining polynomial smaller and often reveals a familiar pattern.

How do I factor x squared plus bx plus c?

Find two values whose product is c and whose sum is b. Those values become the constants in the two factors when the leading coefficient is 1.

How can I tell whether a trinomial is a perfect square?

The first and last terms must be squares, and the middle term must equal positive or negative twice the product of their square roots.

When is a polynomial factored completely?

It is complete over the stated number system when none of its nonconstant factors can be split further in that system. Over the integers, x squared plus 4 does not factor further.

What is the quickest way to check a factorization?

Multiply the factors and compare the expanded coefficients term by term. Testing one numerical value can catch some errors, but expansion verifies the full identity.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.