Factoring Special Products

MATH 018

Combined Algebra

S. Rook

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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

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Notion of a Perfect Square

• A number n is a perfect square if we can find an Integer k such that k · k = n– i.e. the same Integer times itself and k is the

square root of n– e.g.: 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– e.g.: p4 is a perfect square (4 is divisible by 2)

x3 is NOT a perfect square

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Perfect Square Trinomials

• Remember to ALWAYS look for a GCF before factoring!

• Consider what happens when we FOIL (a + b)2

(a + b)2 = a2 + 2ab + b2

• a2 comes from squaring a in (a + b)2

• 2ab comes from doubling the product of a and b in (a + b)2

• b2 comes from squaring b in (a + b)2

Factoring Perfect Square Trinomials

• To factor a perfect square trinomial (e.g. x2 + 2x + 1), we reverse the process:– Answer the following questions:

• Are BOTH end terms perfect squares?– If yes, let a be the square root of the first term

and b be the square root of the last term• Is the middle term 2 times a and b?

– If the answer to BOTH questions is YES, we can factor a2 + 2ab + b2 as (a + b) (a + b) = (a + b)2

– Otherwise, we must seek a new factoring strategy

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Factoring Perfect Square Trinomials (Continued)

– This is the quick way to factor a perfect square trinomial, but it can also be treated as an easy/hard trinomial

– You should be able to identify whether or not a trinomial is also a perfect square trinomial

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Factoring Perfect Square Trinomials (Example)

Ex 1: Factor completely:

a) x2y2 – 8xy2 + 16y2

b) -4r2 – 4r – 1

c) 4n2 + 12n + 9

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Factoring the Sum & Difference of Two Squares

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Difference of Two Squares

• Remember to ALWAYS look for a GCF before factoring!

• A binomial is considered a Difference of Two Squares when BOTH terms are perfect squares separated by a minus sign (e.g. x2 – 1)

• Consider what happens when we FOIL (a + b)(a – b)

a2 comes from the F term in (a + b)(a – b)

b2 comes from the L term in (a + b)(a – b)

Factoring a Difference of Two Squares

• To factor a difference of two squares (e.g. x2 – 1), we reverse the process:– Answer the following questions:

• Are both terms a2 and b2 perfect squares of a and b respectively?

• Is there a minus sign between a2 and b2?

– If the answer to BOTH questions is YES, a2 – b2 can be factored to (a + b)(a – b)

– Otherwise, the polynomial is not a difference of two squares

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Factoring the Difference of Two Squares (Example)

Ex 3: Factor completely:

a) x2 – 64y2

b) 6z2 – 54

c) 2x2 + 128

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Factoring the Difference & Sum of Two Cubes

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Sum & Difference of Two Cubes

• Remember to ALWAYS look for a GCF before factoring!

• Consider multiplying (a + b)(a2 – ab + b2)a3 + b3

• In a similar manner, multiplying (a – b)(a2 + ab + b2) = a3 – b3

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Sum & Difference of Two Cubes

• Thus: a3 + b3 = (a + b)(a2 – ab + b2)

a3 – b3 = (a – b)(a2 + ab + b2)

a3 (+/ –) b3 = (a b)(a ab + b2)

|__same__| |

|__opposite____|

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 a3 and b3 perfect cubes?

– If the answer is YES, a3 – b3 or a3 + b3 can be factored into (a – b)(a2 + ab + b2) or (a + b)(a2 – ab + b2) respectively

– Otherwise, the polynomial is prime

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Factoring the Sum & Difference of Two Cubes (Example)

Ex 4: Factor completely:

a) x3 – 8

b) 27y3 + 64z3

c) 250r3 – 2s3

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Factoring Completely

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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 or 7 cannot be factored further

Factoring Completely (Example)

Ex 5: Factor completely:

a) x4 – 1

b) y4 – 16z4

c) r4t – s4t

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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)