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Factoring Special Quadratic Polynomials

WHAT TO DO: HOW TO DO IT:
⇒ Check for common factors and factor them out of the polynomial.

 

25ax3 - 40 abx2 + 15a2x2

5 ax2(5x - 8b + 3a)

If there is no common factor check for the two special types of factorable polynomials:

 

a) Difference of Squares (binomial)

b) Perfect Square Trinomial

(a) Difference of Squares (binomial)

The difference of squares always factors to the sum and difference of the square roots of those squares.

A2 − B2 = (A + B)(A − B)
i) Factor 4x2 − 9

Difference of Squares (binomial)

i) 4x2 − 9 =

(2x + 3)(2x − 3)

ii) Factor 9x2 − 25

Difference of Squares (binomial)

ii) 9x2 − 25 =

(3x + 5)(3x − 5)

b) Perfect Square Trinomial
Perfect square trinomials must have the first and last terms be perfect squares and the last sign positive. 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.
The "middle sign" is the "sign of the binomial".
∴ Factor the trinomial: 25x2 + 60x + 36 = (5x + 6) 2
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.
 
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