Factoring rewrites a sum or difference as a product. In this lesson, the core tools are extracting the greatest common factor and grouping four terms so the two groups share an identical binomial factor.
Learning objectives
- Extract the complete numeric and variable GCF.
- Choose a negative GCF when it creates a clearer leading term.
- Factor four terms by grouping.
- Normalize opposite binomial factors.
- Verify every proposed factorization by distribution.
Factor the greatest common factor
Reverse the Distributive Property
A five-step factoring flow that identifies the GCF, rewrites every term, factors, and verifies by distribution.
- Polynomial\[18x^3y-30x^2y^2\]
Start with every term visible.
- Identify the GCF\[\operatorname{GCF}=6x^2y\]
Use the numeric GCF and the smallest shared variable powers.
- Rewrite each term\[6x^2y(3x)-6x^2y(5y)\]
The GCF must divide every term exactly.
- Factor outside\[6x^2y(3x-5y)\]
Keep every remaining term inside the parentheses.
- Check by distribution\[6x^2y(3x-5y)=18x^3y-30x^2y^2\]
Expanding reproduces the original polynomial.
Extract a negative GCF
Factor \(-20x^4+30x^3-10x^2\).
- Greatest shared factor
The coefficients share \(10\), and the smallest variable power is \(x^2\).
- Choose the sign
Extract \(-10x^2\) so the leading term inside is positive.
- Divide every term
The quotients are \(2x^2,-3x,1\).
Factor by grouping
Factoring by grouping
Bracket labels show two term groups, each group's common factor, and the repeated binomial.
\[6x^2+9x+4xy+6y\]
Terms 1 and 2
\(6x^2+9x\)
\(3x(2x+3)\)
Terms 3 and 4
\(4xy+6y\)
\(2y(2x+3)\)
Group four terms
- Pair
Group the first two and last two terms; reorder only when needed and mathematically valid.
- Factor each pair
Extract each pair's GCF.
- Normalize signs
Use \(a-b=-(b-a)\) if the binomial factors are opposites.
- Factor again
Extract the now-common binomial.
- Verify
Expand the product to reproduce all original terms and signs.
Factor by grouping
Factor \(8x^2-12x+10xy-15y\).
- \((2x-3)(4x+5y)\)
- \((2x+3)(4x-5y)\)
- \((4x-3)(2x+5y)\)
Show answer and explanation
Answer: \((2x-3)(4x+5y)\)
Group to get \(4x(2x-3)+5y(2x-3)\), then extract \(2x-3\).
Key takeaways
What to remember
- Extract the complete GCF.
- A negative GCF can simplify signs.
- Grouping requires identical binomial factors.
- Expand to verify every factorization.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.