Unit 10.2 Factoring Trinomials and Special Cases
Difference of Squares
The difference of squares is a specific algebraic identity that states that the difference between the squares of two numbers can be factored into a product of two binomials. It is expressed as
Difference of Squares
a2 – b2 = (a + b)(a – b)
For the difference of squares, it must be a minus sign and should not be confused with (a2 + b2) which cannot be factored in this method
Factoring using difference of squares
factor x2 – 9
9 is a perfect square of 3 so we can rewrite it as
x2 – 32
which matches the a2 – b2 format. We are able to factor it
(x – 3)(x + 3)
Sum and Difference of Cubes
The sum and difference of cubes are specific algebraic identities that help in factoring expressions of the form (a3 + b3) or (a3 – b3)
Sum of Cubes
a3 + b3 = (a + b) (a2 – ab + b2)
Difference of Cubes
a3 – b3 = (a – b) (a2 + ab + b2)
Factoring using difference of cubes
Factor 27y3 – 1
27 , y3 and 1 are all perfect cubes so we are able to rewrite it as
(3y)3 – (1)3
Which matches the a3 – b3 format. We are able to factor it using a3 – b3 = (a – b) (a2 + ab + b2)
(3y)3 – (1)3 = (3y – 1) ((3y)2 + (3y)(1) + 12)
= (3y – 1) (9y2 + 3y + 1)