Any questions on the Section 5.6 homework?PowerPoint PresentationSection 5.7Factoring by Special ProductsSlide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Review of techniques from sections 5.5-5.7: Choosing a Factoring StrategySlide 18Slide 19Slide 20Reminder:Slide 22Any questions on the Section 5.6 homework?Now pleaseCLOSE YOUR LAPTOPSand turn off and put away your cell phones.Sample Problems Page Link(Dr. Bruce Johnston)Section 5.7Factoring by Special ProductsPreviously, we learned a shortcut for squaring a binomial.(a + b)2 = a2 + 2ab + b2(a – b)2 = a2 – 2ab + b2So if the first and last terms of our polynomial to be factored can be written as expressions squared, and the middle term of our polynomial is twice the product of those two expressions, then we can use these two previous equations to easily factor the polynomial.a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2Factoring by Special ProductsFactor the polynomial 16x2 – 8xy + y2.Since the first term, 16x2, can be written as (4x)2, and the last term, y2 is obviously a square, we check the middle term.8xy = 2(4x)(y) (twice the product of the expressions that are squared to get the first and last terms of the polynomial) Therefore 16x2 – 8xy + y2 = (4x – y)2.Note: You can use FOIL method to verify that the factorization for the polynomial is accurate. (Multiply (4x – y)(4x – y) and show you get 16x2 – 8xy + y2 .)Example from the online homework:Note that this would be a pretty tough one to do by the “factoring by grouping” method, since 36×121=HUGE NUMBER.But if we notice that 36 = 62 and 121=112, then we can check and see if(6x + 11)2 might be the correct factoring. And sure enough, when we multiply out (6x + 11)(6x + 11), we do get the correct trinomial after we simplify: 36x2 + 66x + 66x + 11 = 36x2 + 132x + 121.Previously, we also discovered a formula for finding the product of the sum and difference of two terms (a – b)(a + b) = a2 – b2 We can use the reverse of the previous equation to see how to quickly factor the difference of 2 squares. a2 – b2 = (a – b)(a + b)This formula can really save you some time!Factor x2 – 16.Since this polynomial can be written as x2 – 42, x2 – 16 = (x – 4)(x + 4).Factor 9x2 – 4.Since this polynomial can be written as (3x)2 – 22,9x2 – 4 = (3x – 2)(3x + 2).Factor 16x2 – 9y2.Since this polynomial can be written as (4x)2 – (3y)2, 16x2 – 9y2 = (4x – 3y)(4x + 3y).Factor x8 – y6.Since this polynomial can be written as (x4)2 – (y3)2, x8 – y6 = (x4 – y3)(x4 + y3).Factor x2 + 4.This one is the sum of two squares, not the difference of squares, so it can’t be factored. This polynomial is a prime polynomial.Remember that you should always factor out any common factors first, before you start any other technique.Step 1: Factor out the GCF, which in this case is 4. 36x2 – 64 = 4(9x2 – 16)Step 2: Factor the polynomial 9x2 – 16The polynomial can be written as (3x)2 – (4)2, so (9x2 – 16) = (3x – 4)(3x + 4). Our final result is 36x2 – 64 = 4(3x – 4)(3x + 4). ↑ (Don’t forget to write in 4 (the GCF) as part of your final answer!)Factor 36x2 – 64.Example from the online homework:Note that 4096 is a HUGE NUMBER. But if it is a square, then we’re in business. How do you tell if it’s a perfect square? Take the square root (on your calculator…)Answer: 4096 = 642. So s4 – 4096 = (s2)2 – 642 = (s2 – 64)(s2 + 64)Are we done yet? No, because s2 – 64 factors further into (s + 8)(s – 8)Final answer: (s + 8)(s – 8)(s2 + 64)Factor x4 – 13x2 + 36. (Make sure you always factor completely!)• x4 is (x2)2, so our factors will look like (x2 + __ )(x2 + __ )•Step 1: Find two factors of 36 that add up to -13 Answer: -9 and -4• So our factors are(x2 – 9)(x2 – 4) (Note that both of the factors are differences of squares, so you’re not done yet!)• So this further factors into(x – 3)(x + 3)(x – 2)(x + 2)There are two additional types of binomials that can be factored easily by remembering a formula. We have not studied these special products previously, as they involve cubes of terms, rather than just squares.a3 + b3 = (a + b)(a2 – ab + b2)a3 – b3 = (a – b)(a2 + ab + b2)NOTE: These formulas will be on the pink formula sheet that we hand out at each test and quiz, so you don’t have to memorize them, but you do need to know how to apply them.)(If you don’t yet have a copy of this formula sheet in your notebook, raise your hand and we’ll give you a yellow copy to keep.)1. Factor x3 + 1.Since this polynomial can be written as x3 + 13,we can use the sum of cubes formula, with a = x and b = 1Answer: x3 + 1 = (x + 1)(x2 – x + 1).2. Factor y3 – 64.Since this polynomial can be written as y3 – 43, we can use the difference of cubes formula with a = y and b = 4.Answer: y3 – 64 = (y – 4)(y2 + 4y + 16). Formulas:Sum of cubes: a3 + b3 = (a + b)(a2 – ab + b2)Diff. of cubes:a3 – b3 = (a – b)(a2 + ab + b2)3. Factor 8t3 + s6.This polynomial can be written as (2t)3 + (s2)3,8t3 + s6 = (2t + s2)((2t)2 – (2t)(s2) + (s2)2) = (2t + s2)(4t2 – 2s2t + s4).4. Factor x3y6 – 27z3.This polynomial can be written as (xy2)3 – (3z)3,x3y6 – 27z3 = (xy2 – 3z)((xy2)2 + (3z)(xy2) + (3z)2) = (xy2 – 3z)(x2y4 + 3xy2z + 9z2).a3 + b3 = (a + b)(a2 – ab + b2)a3 – b3 = (a – b)(a2 + ab + b2)Remember to ALWAYS check to see if you can factor out any common factors before attempting to use any other factoring techniques or formulas.Step 1: Factor out the GCF. (Tip: Since 375 is such a big number, start by factoring the smaller number 24, then see if any of its factors will divide into 375. You will find that the number 3 is a divisor of both 24 and 375.)375y6 – 24y3 = 3y3(125y3 – 8)Since the second part can be written as (5y)3 – 23, 125y3 – 8 = (5y – 2)((5y)2 + (5y)(2) + 22) = (5y – 2)(25y2 + 10y + 4).Final answer: 3y3(5y – 2)(25y2 + 10y + 4).Factor 375y6 – 24y3.Review of techniques from sections 5.5-5.7:Choosing a Factoring StrategySteps for factoring a polynomial:1) Factor out any common factors. (Always check this first, before doing any other factoring method.)2) Look at number of terms in polynomial•If 2 terms, look for difference of squares, difference of cubes or sum …