Please use this form if you would like to have this math solver on your website, free of charge.
Factoring Special Quadratic Polynomials
WHAT TO DO:
HOW TO DO IT:
If there is no common factor check for the two
special types of factorable polynomials:
a) difference of squares
b) perfect square trinomial
(a) difference of squares
The difference of squares always factors to the sum and difference of the square roots of those squares.
A2
− B2 = (A + B)(A − B)
Sometimes the factors themselves contain another
factorable binomial -- difference of squares.
i) (s 4 − t 4)
= (s2 + t2)(s2 − t2)
= (s2 + t2)(s + t)(s − t)
Check for factorable binomial -- difference of squares.
Continue factoring to prime factors:
ii) x8 − y8 =
(x4 + y4)(x4 - y4)
= (x4 + y4)(x2 + y2)(x2
- y2)
= (x4 + y4)(x2 + y2)(x +
y)(x − y)
Factor to prime factors: iii) x16 - y16 = (x8
+ y8)(x4 + y4)(x2 + y2)(x
+ y)(x - y)
b) perfect square trinomial
Perfect square trinomials must have the first and last
terms be perfect squares and the last sign positive. If
all of these conditions hold, check to see if the product
of the square roots of the first term and the last term is
the same as half the middle term or if the middle term
is twice the cross product of the square roots.
i) 4x2
- 12x + 9 = (2x - 3)2
ii) 9x2 + 30x + 25 = (3x + 5)2
iii) 25x2 + 60x + 36 = (5x + 6)2
NOTE: if the last sign is not + or if the middle term
is not twice the square root factors, the trinomial is not
a perfect square trinomial and all must be examined
by other criteria, such as the Grouping Number.
a) 4x2 + 12x - 9
b) 4x2 - 13x + 9
c) 4x2 + 15x + 9
NOTE: If the trinomial isn’t immediately recognized as a perfect square trinomial,
the best method is to treat it as “any trinomial†and use factor by grouping.
