2. Dori’s list on how to factor Polynomial LOOK FOR A GCF or GMF! Factor as product of Binomials Not Factorable Special Product Factor of Grouping Difference of Squares PST
3. How to find GMF Example:5𝑥3and 10𝑥2 So the GMF is… You can figure out the GMF by writing it out, using Prime Factorization =5×𝑥 ×𝑥 ×𝑥 =2×5×𝑥×𝑥 G𝑀𝐹=5𝑥2
4. Factoring Polynomials Like always find the GMF first. In this case the GMF is 4𝑥3 Next, divide each term by the GMF. SO that way each term will be the product of the GCF. Example:8𝑥4− 12𝑥3 GMF: 4𝑥3 8𝑥44𝑥3−12𝑥34𝑥3 =4𝑥3(2𝑥−3) The 8𝑥4 became 2𝑥 and 12𝑥3became 3 after being divided by 4𝑥3
5. What do these to group have in common? Use FOIL to double check your work Before you do anything, group monomials that have a GCF. Look they both have(𝑥3−1)so now all you have to do is factor it out! Factor by Grouping Example: 15𝑥4−15𝑥+12𝑥3−12 (15𝑥4−15𝑥)+(12𝑥3−12) 15𝑥(𝑥3−1)+12(𝑥3−1) (𝑥3−1)(15𝑥+12) 𝑥3−115𝑥+12=15𝑥4−15𝑥+12𝑥3−12
6. What if they are opposites? Example: 5𝑥2−10𝑥+6𝑥−3𝑥2 Like before, group the terms the monomials that have something in common. Find the GMF of each group! Oh my! Look (x-2) and (2-x) are two different things. So they don’t have a common factor? NO! Actually they do, multiply (2-x) by -1 to change it around! Now factor it out, and use FOIL to double check your work! (-2+x) is the same as (x-2). (5𝑥2−10𝑥)+(6𝑥−3𝑥2) 5𝑥(𝑥−2)+3𝑥(2−𝑥) 5𝑥(𝑥−2)+3𝑥(−1)(2−𝑥) 5𝑥𝑥−2−3𝑥−2+𝑥 5𝑥𝑥−2−3𝑥(𝑥−2) (𝑥−2)(5𝑥−3) (𝑥−2)(5𝑥−3)= 5𝑥2−10𝑥+6𝑥−3𝑥2
7. Factoring Binomails: 𝑥2+𝑏𝑥+𝑐 You must find a pair of number, when added equals to B and gives a product of C. (𝑥,𝑦) 𝑡𝑤=𝑐 𝑡 𝑤 t+w=b
8. Product of Binomials: 𝑥2+𝑏𝑥+𝑐 What are some factors of 20 that add up to 9? Look, the group (4,5) add up to 9 and have a product of 20. The first term is 𝑥2 , so that’s why the variables need to have a coefficient of 1. Or (𝑥+∎) (𝑥+∎) Use FOIL to double check you work! Example:𝑥2+9𝑥+20 20=1,20, 2,10, 4,5 (𝑥+4)(𝑥+5) (𝑥+4)(𝑥+5)=𝑥2+9𝑥+20
9. What if 𝑐 is negative? It looks like the pair (2,-16) satisfies both these requirements! Well, this is almost the same, what pair of numbers gives a product a -32, and a sum of -14? Like before because 𝑥2 is the first term, the variable terms need to have a coefficient of 1. Use FOIL to double check you work! Example:𝑥2−14x−32 −32=−1,32, 2,−16, 4,−8, −32,1, (−2,16) (𝑥+2)(𝑥−16) (𝑥+2)(𝑥−16)= 𝑥2−14x−32
10. Product of Binomials: 𝑎𝑥2+𝑏+𝑐 Use the FOIL method, to double check if it works! Hey let’s try the pair (2,5), and see if it works. Remember to use the form (x+∎) (x+∎)! Look here a=3, b=12 and c=10. Try to find a pair of numbers in which there sum is 12 and product is 10. Well this is just guess and check! Split up 3𝑥2 ! Example: 3𝑥2+12x+10 10=1,10, (2,5) (2𝑥+2)(𝑥+5) 2𝑥+2𝑥+5= 3𝑥2+12x+10
11. Perfect Square Trinomial In this case the pair is (-14,-14). Let’s try it out. Use FOIL to see if it works. Yes that’s right, it is equal to(𝑥−14)2! A Perfect Square Trinomial is when you square a binomial quantity. Example: 𝑥2−28𝑥+196 Like before find a pair of numbers that has a sum of -28 and a product of 196 IT WORKS! But do you notice that (x-14) and (x-14) is the same? So that means it is equivalent to … (𝑥−14)(𝑥−14) (𝑥−14)(𝑥−14)=𝑥2−14𝑥−14𝑥+196 (𝑥−14)2
12. Difference of Squares The -56𝑥2𝑦 cancels out 56𝑥2𝑦 leaving us with 64𝑥4−49𝑦2 Factor out the two monomials! Use FOIL to double check your work! A difference of Square's is when a square number is subtracted from another square number. Example: 64𝑥4−49𝑦2 (8𝑥2+7y)( 8𝑥2−7y) (8𝑥2+7y) 8𝑥2−7y=64𝑥4−56𝑥2𝑦+56𝑥2𝑦−49𝑦2 (8𝑥2+7y)( 8𝑥2−7y)
