Special Factoring

Objectives: To factor the difference
of two squares, to factor Perfect
Square Trinomials, and to factor the
sum or difference of two cubes.

Review of 7.1 & 7.2

ο Define factoring.
ο How do we check our answer after
ο factoring?
ο Factor:
ο A) k² – 11kh + 28h²
ο B) 4x² + 2x - 6

Review

ο Multiply using FOIL:
(x – 3) (x + 3).
x² + 3x – 3x - 9

Answer: x² – 9

Remember: the outside/inside terms cancel b/c they
are opposite terms.

ο Multiply: (y + 2)(y – 2)

Answer: y² – 4

ο This answer is a special case called the

DIFFERENCE OF TWO SQUARES
(a.k.a. DOTS).

Factoring the Difference of
Two Squares (DOTS)

ο Clue 1: usually only 2 terms
ο Clue 2: _________ sign
ο Clue 3: _______ terms are perfect squares

ο General Form: x² – y² = (x + y)(x - y)

ο MEMORIZE THESE PERFECT SQUARES:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144,
169, 196, 225, 256,…,
400,…,625,…900,…,10000

Examples – Factor.

ο a) x² – 25
ο b) 3a² – 48
ο c) 169x² – y²
ο d) 81a² – 121b²

Think, pair, share…

ο e) 9x² – z⁴
ο f) 4a⁴ – 49b⁸
ο g) 5y³ – 320y
ο h) (m - 2)² – 100

REVIEW
The Square of a Binomial

ο Multiply (m – 4)².

ο Multiply (5n + 3)².

What to look for…
Ax² + Bx + C

ο Clue 1: A & C are _________, perfect
squares.

ο Clue 2: B is the square root of A times
the square root of C, doubled.

If these two things are true, the trinomial is
a Perfect Square Trinomial (PST) and can
be factored as (x + y)² or (x – y)².

General Form of Perfect
Square Trinomials

ο x² + 2xy + y² = (x + y)²
or
ο x² – 2xy + y² = (x - y)²

ο Note: When factoring, the sign
in the binomial is the _______
as the sign of B in the trinomial.

Just watch and think.

ο Ex) x² + 12x + 36
ο What’s the square root
of A? of C?
ο Multiply these and
double. Does it = B?
ο Then it’s a Perfect
Square Trinomial!

ο Solution: (x + 6)²

ο Ex) 16a² – 56a + 49
ο Square root of A?
of C?
ο Multiply and
double…
ο = B?

ο Solution: (4a – 7) ²

Examples

ο A) x² + 8x + 16
ο B) 9n² + 48n + 64
ο C) 4z² – 36z + 81

Think, pair, share…

ο D) 25c² – 20c + 4 – d²
ο E) 9a² – 24a + 16 – b²

What if the leading term has
an odd power?

ο The problem could be the sum or
difference of two cubes. (page 389)

Factoring the Difference or
Sum of Two Cubes

ο Clue 1: usually only 2 terms
ο Clue 2: both terms are perfect cubes

MEMORIZE THESE PERFECT CUBES:
1, 8, 27, 64, 125, 216,…1000

General Form of Factoring
Cubes

x³ – y³ = (x - y)(x² + xy + y²)
or
x³ + y³ = (x + y)(x² - xy + y²)

Examples

ο A) x³ + 64
ο B) 8n³ – 27
ο C) 125a³ + b³

Think, pair, share…

ο D) 216c³d³ + 1
ο E) 4y³ – 500z³