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Factoring Special Products

2. Overview. Section 6.5 in the textbookFactoring perfect square trinomialsFactoring the sum

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Factoring Special Products

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    1. Factoring Special Products MATH 018 Combined Algebra S. Rook

    2. 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

    3. Factoring Perfect Square Trinomials

    4. 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.: p4 is a perfect square (4 is divisible by 2) x3 is NOT a perfect square

    5. 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 = 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

    6. 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 6

    7. 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

    8. Factoring Perfect Square Trinomials (Example) Ex 1: Factor completely: a) x2y2 – 8xy2 + 16y2 b) -4r2 – 4r – 1 c) 4n2 + 12n + 9 8

    9. Factoring the Sum & Difference of Two Squares

    10. 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. 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)

    11. 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 11

    12. Factoring the Difference of Two Squares (Example) Ex 3: Factor completely: a) x2 – 64y2 b) 6z2 – 54 c) 2x2 + 128 12

    13. Factoring the Difference & Sum of Two Cubes

    14. 14 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

    15. 15 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____|

    16. 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 16

    17. Factoring the Sum & Difference of Two Cubes (Example) Ex 4: Factor completely: a) x3 – 8 b) 27y3 + 64z3 c) 250r3 – 2s3 17

    18. Factoring Completely

    19. 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

    20. Factoring Completely (Example) Ex 5: Factor completely: a) x4 – 1 b) y4 – 16z4 c) r4t – s4t 20

    21. 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)

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