Integrated Math 2 / Unit 2: Polynomials and Factoring
2.6 Special Products
Notes
Some patterns factor quickly.
- Difference of squares: $a^2 - b^2 = (a - b)(a + b)$. Example: $x^2 - 49 = (x - 7)(x + 7)$.
- Perfect square trinomials: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$.
- Example: $4x^2 - 12x + 9 = (2x - 3)^2$ because $2(2x)(3) = 12x$.
- A sum of squares like $x^2 + 9$ does not factor over the real numbers.
- Factor out the GCF first: $2x^3 - 50x = 2x(x - 5)(x + 5)$.
Practice
Factor $16x^2 - 81$.
This is $(4x)^2 - 9^2$, a difference of squares.
Factor $4x^2 - 12x + 9$.
$(2x)^2 = 4x^2$, $3^2 = 9$, and the middle term is $-2(2x)(3) = -12x$.
Factor completely: $2x^3 - 50x$.
Take out $2x$ to get $2x(x^2 - 25)$, then factor the difference of squares.
Ready for more? Take the Unit 2 test.