# Factoring Special Products

**Overview**

• Section 6.5 in the textbook

– Factoring perfect square trinomials

– Factoring the sum & difference of two squares

– Factoring the sum & difference of two cubes

– Factoring completely

**Factoring Perfect Square
Trinomials**

**Notion of a Perfect Square**

• A number n is a perfect square if we can

find an Integer k such that k · k = n

– The same Integer times itself

– Ex: 4 is a perfect square (k = 2)

81 is a perfect square (k = ?)

• A variable is a perfect square if its

exponent is evenly divisible by 2

– Ex:

p^{4} is a perfect square (4 is divisible by 2)

x^{3} is NOT a perfect square

**Perfect Square Trinomials **

• Remember to ALWAYS look for a GCF

before factoring!

• Consider what happens when we FOIL

(a + b)^{2}

(a + b)^{2} = a^{2} + 2ab + b^{2}

• a^{2} comes from squaring a in (a + b)^{2}

• 2ab comes from doubling the product of a

and b in (a + b)^{2}

• b^{2} comes from squaring b in (a + b)^{2}

**Factoring Perfect Square
Trinomials**

• To factor a perfect square trinomial, we

reverse the process:

– Answer the following questions:

• Are BOTH end terms a ^{2} and b^{2} perfect squares of

a and b respectively?

• Is the middle term two times a and b?

– If the answer to BOTH questions is YES, we can

factor a^{2} + 2ab + b^{2} as (a + b) (a + b) = (a + b)^{2}

– Otherwise, we must seek a new factoring

strategy

Ex 1: Factor completely:

a) x^{2}y^{2} – 8xy + 16y^{2}

b) -4r^{2} – 4r – 1

c) 36n^{2} + 36n + 9

**Factoring the Sum &
Difference of Two Square **

**Difference of Two Squares
**• Remember to ALWAYS look for a GCF

before factoring!

• 2 terms both of which are perfect squares

• Consider what happens when we FOIL

(a + b)(a – b)

(a + b)(a – b) = a^{2} – b^{2}

• a^{2}comes from the F term in (a + b)(a – b)

• b^{2}comes from the L term in (a + b)(a – b)

**Factoring a Difference of Two
Squares**

• To factor a difference of two squares, we

reverse the process:

– Answer the following questions:

• Are both terms a^{2} and b^{2} perfect squares of a and

b respectively?

• Is the sign between a^{2} and b^{2} a – ?

– If the answer to BOTH questions is YES,

a^{2} – b^{2} can be factored to (a + b)(a – b)

– Otherwise, the polynomial is prime

Ex 3: Factor completely:

a) x^{2} – 64y^{2}

b) 6z^{2} – 54

• Remember to ALWAYS look for a GCF before

factoring!

• Consider a^{2} + b^{2}

• Only 3 possibilities for the factoring:

(a + b)(a + b) = a^{2} + 2ab + b^{2} ≠ a^{2} + b^{2}

(a – b)(a – b) = a^{2} – 2ab + b^{2} ≠ a^{2} + b^{2}

(a + b)(a – b) = (a – b)(a + b) = a^{2} – b^{2} ≠ a^{2} + b^{2}

• Therefore, the sum of two squares is PRIME

Ex 4: Factor completely:

2x^{2} + 128

**Factoring the Difference &
Sum of Two Cubes**

**Sum & Difference of Two Cubes**

• Remember to ALWAYS look for a GCF

before factoring!

• Consider (a + b)(a^{2} – ab + b^{2})

a(a^{2} – ab + b^{2}) + b(a^{2} – ab + b^{2})

a^{3} – a^{2}b + ab^{2} + a^{2}b – ab^{2} + b^{3}

a^{3} + b^{3}

• Consider (a – b)(a^{2} + ab + b^{2})

a(a^{2} + ab + b^{2}) – b(a^{2} + ab + b^{2})

a^{3} – b^{3}

• Thus:

a^{3} + b^{3} = (a + b)(a^{2} – ab + b^{2})

a^{3} – b^{3} = (a – b)(a^{2} + ab + b^{2})

**Factoring a Sum or Difference
of Two Cubes**

• To factor a sum or difference of two cubes,

we reverse the process:

– Answer the following question:

• Are both terms a^{3} and b^{3} perfect cubes of a and b

respectively?

– If the answer is YES, a^{3} – b^{3} or a^{3} + b^{3} can be

factored into (a – b)(a^{2} + ab + b^{2}) or

(a + b)(a^{2} – ab + b^{2}) respectively

– Otherwise, the polynomial is prime

Ex 5: Factor completely:

a) x^{3} – 8

b) 27y^{3} + 64z^{3}

c) 250r^{3} – 2s^{3}

**Factoring Completely**

• Remember to ALWAYS look for a GCF

before factoring!

• Choose a factoring strategy based on the

number of terms

• Look at the result to see if any of the

products can be factored further

– Polynomials with a degree of 1 or less cannot

be factored further

• e.g. 2x + 1, 7

Ex 6: Factor completely:

a) x^{4} – 1

b) y^{4} – 16z^{4}

c) r^{4}t – s^{4}t

**Summary**

• After studying these slides, you should know

how to do the following:

– Recognize and factor a perfect square trinomial

– Factor a difference of two squares

– Recognize that the sum of two squares is prime

– Factor the difference or sum of two cubes

– Completely factor a polynomial

• Additional Practice

– See the list of suggested problems for 6.5

• Next lesson

– Solving Quadratic Equations by Factoring (Section

6.6)

Prev | Next |