Special products turn certain factoring tasks into recognition problems. The structure must match exactly: a difference of two squares or a three-term perfect-square pattern with the correct middle term.
Three exact identities
| Pattern | Expanded form | Factored form | Recognition check | Common mistake |
|---|---|---|---|---|
| Difference of squares | \(a^2-b^2\) | \((a+b)(a-b)\) | Two terms, subtraction, both perfect squares | Trying to factor \(a^2+b^2\) the same way |
| Perfect square, plus | \(a^2+2ab+b^2\) | \((a+b)^2\) | Middle term equals positive \(2ab\) | Omitting the middle coefficient \(2\) |
| Perfect square, minus | \(a^2-2ab+b^2\) | \((a-b)^2\) | Middle term equals negative \(2ab\) | Changing the last term's sign |
Difference of squares is not a binomial square
Correct
\(a^2-b^2=(a+b)(a-b)\); the middle terms cancel.
Incorrect look-alike
\((a-b)^2=a^2-2ab+b^2\), which contains three terms after expansion.
GCF before the special pattern
- Extract
Find and factor the greatest common factor.
- Reinspect
Check whether the remaining expression matches a special identity.
- Apply
Factor the difference of squares or perfect-square trinomial.
- Finish
Check whether any factor can be factored again.
- Verify
Expand back to the original polynomial.
GCF, then difference of squares
Factor \(18x^3-72x\) completely.
- GCF
\(18x^3-72x=18x(x^2-4)\).
- Recognize
\(x^2-4=x^2-2^2\) is a difference of squares.
- Factor
\(x^2-4=(x+2)(x-2)\).
Zero Product Property
Solve by factoring
Solve \(4x^2-49=0\).
- Factor
\((2x+7)(2x-7)=0\).
- Use zero product
\(2x+7=0\) or \(2x-7=0\).
- Solve both
\(x=-\frac72\) or \(x=\frac72\).
Use identities without unnecessary expansion
Common mistakes
- \(a^2+b^2\) is not a difference of squares over the real numbers.
- \(a^2-b^2\) does not equal \((a-b)^2\).
- A perfect-square trinomial requires the exact middle term \(\pm2ab\).
- Do not use the Zero Product Property until the equation equals \(0\).
- Keep both roots when two linear factors equal zero.
Recognize a perfect square
Factor \(9x^2-30x+25\).
- \((3x-5)^2\)
- \((3x+5)^2\)
- \((9x-5)(x-5)\)
- \((3x-5)(3x+5)\)
Show answer and explanation
Answer: \((3x-5)^2\)
The outer squares are \((3x)^2\) and \(5^2\), and the middle term is \(-2(3x)(5)=-30x\).
Key takeaways
What to remember
- Difference of squares requires subtraction and two perfect-square terms.
- Perfect-square trinomials require the exact middle term \(\pm2ab\).
- Always factor the GCF before applying a special pattern.
- Set the equation equal to zero before using the Zero Product Property.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.