Try our Free Online Math Solver!

Factoring Special Products
Perfect Square Trinomials
Perfect Square Trinomials
a^{2} + 2ab + b^{2} = (a + b)^{2} 
In order for a polynomial to be a perfect square
trinomial , two conditions must be satisfied:
1. The first and last terms must be perfect squares.
2. The “ middle term ” must equal 2 or – 2 times the
product of the expressions being squared in the
first and last term.
Example: Factor the following :
x^{2} + 8x + 16 = (x + 4)^{2}
b.) 9x^{4} − 30x^{2}z + 25z^{2}
9x^{4} − 30x^{2}z + 25z^{2} = (3x^{2} – 5z)^{2}
Difference of Two Squares a^{2} – b^{2} = (a + b)(a − b) 
Example: Factor the following :
a.) 3x^{2} − 27
3x^{2} − 27 = 3(x^{2} – 9) 3 is a common factor.
= 3(x + 3)(x – 3)
b.) 4x^{2} − 25y^{4}
4x^{2} − 25y^{4} = (2x + 5y^{2})(2x – 5y^{2})
The Sum of Two Cubes
The Sum of Two Cubes a^{3} + b^{3} = (a + b) (a^{2} − ab + b^{2}) 
Example: Factor: 27x^{3} + 125
a = 3x
b = 5
27x^{3} + 125 = (3x)^{3} + 5^{3}
= (3x + 5) [(3x)^{2} − 3x·5 + 5^{2}]
= (3x + 5)(9x^{2} − 15x + 25)
The Difference of Two Cubes
The Difference of Two Cubes a^{3} − b^{3} = (a − b) (a^{2} + ab + b^{2}) 
Example: Factor: x^{3} − 64
a = x
b = 4
x^{3} − 64 = x^{3} – 4^{3}
= (x − 4) (x^{2} + 4x + 4^{2})
= (x − 4) (x^{2} + 4x + 16)
Practice
Factor each perfect square trinomial completely
x^{2} + 14x + 49
x^{2} − 8x + 16
4a^{2} − 12ab + 9b^{2}
4x^{2} + 20x + 25
n^{2} − 16
a^{4} − 16
36 p^{2} − 49q^{2}
9x^{6} − 25y^{8}
64x^{3} + 1
8x^{3} − 27
4x^{4} − 4xy^{3}
125a^{6} + 8b^{3}
Prev  Next 