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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

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

A^{2} − B^{2}
= (A + B)(A − B)

a) Factor 4x^{2} − 9 difference of squares (binomial)

a) 4x^{2}
− 9 = (2x + 3)(2x − 3)

b) Factor 9x^{2} − 25 difference of squares (binomial)

b) 9x^{2}
− 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. 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.

The â€œmiddle signâ€
is the â€œsign of the binomialâ€.

→ Factor the trinomial: 25x^{2} + 60x + 36

= (5x + 6)^{2}

NOTE: If the trinomial isnâ€™t immediately recognized as a perfect square trinomial,
the best method is to treat it as â€œany trinomialâ€ and use factor by grouping.