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Factoring the Special Products

- There are 2 special products that you need to be

able to factor by recognition, - (a) the difference of 2 squares and
- (b) perfect square trinomials

Recognition the difference of 2 squares is

easiest to recognize because the polynomial has

to be a binomial with each of the 2 terms being a

perfect square and having a minus sign between

them.

Variables are perfect squares if their exponents

are even numbers. Examples

- Numbers are perfect squares if they are the

answer to squaring a whole number. - Examples1, 4, 9, 16, 25, 36, 49, 64, 81, 100,

121, 144, 169, etc.

The difference of 2 squares always factors into

(sq.rt 1st term sq.rt 2nd term) times (sq.rt

1st term - sq.rt 2nd term)or

Note The sum of 2 squares will not factor!!

- The Difference of Squares

Difference of Squares

x2 y2 (x y)(x y)

Factoring Differences of Squares

- AMPLE 1

Factor each polynomial.

(a) 2n2 50

There is a common factor of 2.

Factor out the common factor. 2n2

50 2(n2 25)

2(n 5)(n 5) Factor the

difference of squares.

Factoring Differences of Squares

Factor each polynomial.

(b) 9g2 16

9g2 16

(3g)2 (4)2

(3g 4)(3g 4)

A2

(A B)(A B)

B2

- Caution

CAUTION

Assuming no greatest common factor except 1, it

is not possible to factor (with real numbers) a

sum of squares, such as x2 16.

Factor

RecognitionPerfect square trinomials will have

1st and 3rd terms that are perfect squares and

the middle term will be twice the product of the

square roots of the 1st and 3rd terms. The 1st

and 3rd term will always be positive and the

middle term can be positive or negative.

Be sure that the trinomial is a perfect square

trinomial before factoring. Which are perfect

square trinomials?

A perfect square trinomial will always

factor(sqrt of 1st term sign sqrt of 2nd term)

where sign is the same as the sign of the 2nd

term of the trinomial.

Factor

Factor completely

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