Factoring Special Products MATH 018 Combined Algebra S. Rook

2 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

4 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.: p 4 is a perfect square (4 is divisible by 2) x 3 is NOT a perfect square

5 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 (e.g. x 2 + 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 a 2 + 2ab + b 2 as (a + b) (a + b) = (a + b) 2 –Otherwise, we must seek a new factoring strategy 6

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 7

Factoring Perfect Square Trinomials (Example) Ex 1: Factor completely: a) x 2 y 2 – 8xy y 2 b) -4r 2 – 4r – 1 c) 4n n + 9 8

Factoring the Sum & Difference of Two Squares

10 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. x 2 – 1) Consider what happens when we FOIL (a + b)(a – b) 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 (e.g. x 2 – 1), we reverse the process: –Answer the following questions: Are both terms a 2 and b 2 perfect squares of a and b respectively? Is there a minus sign between a 2 and b 2 ? –If the answer to BOTH questions is YES, a 2 – b 2 can be factored to (a + b)(a – b) –Otherwise, the polynomial is not a difference of two squares 11

Factoring the Difference of Two Squares (Example) Ex 3: Factor completely: a) x 2 – 64y 2 b) 6z 2 – 54 c)2x

Factoring the Difference & Sum of Two Cubes

14 Sum & Difference of Two Cubes Remember to ALWAYS look for a GCF before factoring! Consider multiplying (a + b)(a 2 – ab + b 2 ) a 3 + b 3 In a similar manner, multiplying (a – b)(a 2 + ab + b 2 ) = a 3 – b 3

15 Sum & Difference of Two Cubes Thus: a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) a 3 (+/ –) b 3 = (a b)(a ab + b 2 ) |__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 a 3 and b 3 perfect cubes? –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 16

Factoring the Sum & Difference of Two Cubes (Example) Ex 4: Factor completely: a) x 3 – 8 b) 27y z 3 c)250r 3 – 2s 3 17

Factoring Completely

19 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) x 4 – 1 b) y 4 – 16z 4 c) r 4 t – s 4 t 20

21 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)