Recognize and factor expressions of the form a² − b² = (a+b)(a−b), with step-by-step identification, FOIL verification, and repeated factoring for higher-degree expressions.
x^2 - 25, 4x^2 - 9, or 16x^4 - 1. Use ^ for exponents.2x.a² − b² = (a + b)(a − b)
A difference of squares is any expression of the form a² − b², where both terms are perfect squares and they are subtracted (differenced).
Three conditions must all be true:
Geometric interpretation: Imagine a large square of area a² with a smaller square b² cut from its corner. The remaining L-shaped region has area a² − b², which can be rearranged into a rectangle of dimensions (a+b) by (a−b).
1. Sum of squares does NOT factor
a² + b² cannot be factored over the real numbers. There is no real factoring for x² + 25. Students often incorrectly write (x+5)² — that is x² + 10x + 25, not x² + 25.
2. The degree must be even
x³ − 25 does not fit the pattern because x³ is not a perfect square. Both terms need even exponents (0, 2, 4, …).
3. Don't forget to check for repeated factoring
After factoring 16x⁴ − 1 = (4x²+1)(4x²−1), notice that 4x²−1 is itself a difference of squares: (2x+1)(2x−1)!
One-on-one Algebra 1 tutoring makes difference of squares, trinomials, and all factoring patterns click — we work through your exact problems and build pattern recognition that sticks for tests.