Any questions on the Section 5 6 homework Now please CLOSE YOUR LAPTOPS and turn off and put away your cell phones Sample Problems Page Link Dr Bruce Johnston Section 5 7 Factoring by Special Products Factoring by Special Products Previously we learned a shortcut for squaring a binomial a b 2 a2 2ab b2 a b 2 a2 2ab b2 So 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 2 Factor 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 Factor 36x2 64 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 16 The 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 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 Formulas Sum of cubes a3 b3 a b a2 ab b2 Diff of cubes a3 b3 a b a2 ab b2 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 1 Answer 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 a3 b3 a b a2 ab b2 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 Remember to ALWAYS check to see if you can factor out any common factors before attempting to use any other factoring techniques or formulas Factor 375y6 24y3 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 Review of techniques from sections 5 5 5 7 Choosing a Factoring Strategy Steps 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 of cubes Use the formula sheet for these If 3 terms use techniques for factoring into 2 binomials If 4 or more terms try factoring by grouping 3 See if any factors can be further factored 4 Check by multiplying all factors out to make sure you get back to the original polynomial Factor x2y y3 Factor out the GCF x2y y3 y x2 y2 Since the remaining polynomial …