Factoring polynomials


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Factoring polynomials

  1. 1. Factoring Polynomials<br />Ann Georgy and Tiffany Kwok<br />Period 6<br />
  2. 2. Dori’s list on how to factor<br />Polynomial<br />LOOK FOR A GCF or GMF!<br />Factor as product of Binomials<br />Not Factorable<br />Special Product<br />Factor of Grouping<br />Difference of Squares <br />PST<br />
  3. 3. How to find GMF<br />Example:5π‘₯3and 10π‘₯2<br />Β <br />So the GMF is…<br />You can figure out the GMF by writing it out, using Prime Factorization<br />=5Γ—π‘₯Β Γ—π‘₯Β Γ—π‘₯<br />Β <br />=2Γ—5Γ—π‘₯Γ—π‘₯<br />Β <br />G𝑀𝐹=5π‘₯2<br />Β <br />
  4. 4. Factoring Polynomials<br />Like always find the GMF first. In this case the GMF is 4π‘₯3<br />Β <br />Next, divide each term by the GMF. SO that way each term will be the product of the GCF.<br />Example:8π‘₯4βˆ’Β 12π‘₯3<br />Β <br />GMF: 4π‘₯3<br />Β <br />8π‘₯44π‘₯3βˆ’12π‘₯34π‘₯3Β <br />Β <br />=4π‘₯3(2π‘₯βˆ’3)<br />Β <br />The 8π‘₯4Β became 2π‘₯Β and 12π‘₯3became 3 after being divided by 4π‘₯3<br />Β <br />
  5. 5. What do these to group have in common?<br />Use FOIL to double check your work<br />Before you do anything, group monomials that have a GCF.<br />Look they both have(π‘₯3βˆ’1)so now all you have to do is factor it out!<br />Β <br />Factor by Grouping<br />Example: 15π‘₯4βˆ’15π‘₯+12π‘₯3βˆ’12<br />Β <br />(15π‘₯4βˆ’15π‘₯)+(12π‘₯3βˆ’12)<br />Β <br />15π‘₯(π‘₯3βˆ’1)+12(π‘₯3βˆ’1)<br />Β <br />(π‘₯3βˆ’1)(15π‘₯+12)<br />Β <br />π‘₯3βˆ’115π‘₯+12=15π‘₯4βˆ’15π‘₯+12π‘₯3βˆ’12<br />Β <br />
  6. 6. What if they are opposites?<br />Example: 5π‘₯2βˆ’10π‘₯+6π‘₯βˆ’3π‘₯2<br />Β <br />Like before, group the terms the monomials that have something in common.<br />Find the GMF of each group!<br />Oh my! Look (x-2) and (2-x) are two different things. So they don’t have a common factor? <br />NO! Actually they do, multiply (2-x) by -1 to change it around!<br />Now factor it out, and use FOIL to double check your work!<br />(-2+x) is the same as (x-2).<br />(5π‘₯2βˆ’10π‘₯)+(6π‘₯βˆ’3π‘₯2)<br />Β <br />5π‘₯(π‘₯βˆ’2)+3π‘₯(2βˆ’π‘₯)<br />Β <br />5π‘₯(π‘₯βˆ’2)+3π‘₯(βˆ’1)(2βˆ’π‘₯)<br />5π‘₯π‘₯βˆ’2βˆ’3π‘₯βˆ’2+π‘₯<br />5π‘₯π‘₯βˆ’2βˆ’3π‘₯(π‘₯βˆ’2)<br />(π‘₯βˆ’2)(5π‘₯βˆ’3)<br />Β <br />(π‘₯βˆ’2)(5π‘₯βˆ’3)= 5π‘₯2βˆ’10π‘₯+6π‘₯βˆ’3π‘₯2<br />Β <br />
  7. 7. Factoring Binomails: π‘₯2+𝑏π‘₯+𝑐<br />Β <br />You must find a pair of number, when added equals to B and gives a product of C.<br />(π‘₯,𝑦)<br />Β <br />𝑑𝑀=𝑐<br />Β <br />𝑑<br />Β <br />𝑀<br />Β <br />t+w=b<br />Β <br />
  8. 8. Product of Binomials: π‘₯2+𝑏π‘₯+𝑐<br />Β <br />What are some factors of 20 that add up to 9?<br />Look, the group (4,5) add up to 9 and have a product of 20.<br />The first term is π‘₯2Β , so that’s why the variables need to have a coefficient of 1. Or (π‘₯+∎)Β (π‘₯+∎)<br />Β <br />Use FOIL to double check you work!<br />Example:π‘₯2+9π‘₯+20<br />20=1,20,Β 2,10,Β 4,5<br />Β <br />(π‘₯+4)(π‘₯+5)<br />Β <br />(π‘₯+4)(π‘₯+5)=π‘₯2+9π‘₯+20<br />Β <br />
  9. 9. What if 𝑐 is negative?<br />Β <br />It looks like the pair (2,-16) satisfies both these requirements!<br />Well, this is almost the same, what pair of numbers gives a product a -32, and a sum of -14?<br />Like before because π‘₯2Β  is the first term, the variable terms need to have a coefficient of 1.<br />Β <br />Use FOIL to double check you work!<br />Example:π‘₯2βˆ’14xβˆ’32<br />Β <br />βˆ’32=βˆ’1,32,Β 2,βˆ’16,Β 4,βˆ’8,Β βˆ’32,1,Β (βˆ’2,16)Β <br />Β <br />(π‘₯+2)(π‘₯βˆ’16)<br />Β <br />(π‘₯+2)(π‘₯βˆ’16)= π‘₯2βˆ’14xβˆ’32<br />Β <br />
  10. 10. Product of Binomials: π‘Žπ‘₯2+𝑏+𝑐<br />Β <br />Use the FOIL method, to double check if it works!<br />Hey let’s try the pair (2,5), and see if it works. Remember to use the form (x+∎) (x+∎)!<br />Β <br />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!<br />Split upΒ 3π‘₯2 ! <br />Β <br />Example: 3π‘₯2+12x+10<br />Β <br />10=1,10,Β (2,5)<br />Β <br />(2π‘₯+2)(π‘₯+5)<br />Β <br />2π‘₯+2π‘₯+5=<br />3π‘₯2+12x+10<br />Β <br />
  11. 11. Perfect Square Trinomial<br />In this case the pair is (-14,-14). Let’s try it out.<br />Use FOIL to see if it works.<br />Yes that’s right, it is equal to(π‘₯βˆ’14)2! <br />Β <br />A Perfect Square Trinomial is when you square a binomial quantity. <br />Example: π‘₯2βˆ’28π‘₯+196Β <br />Β <br />Like before find a pair of numbers that has a sum of -28 and a product of 196<br />IT WORKS! But do you notice that (x-14) and (x-14) is the same? So that means it is equivalent to …<br />(π‘₯βˆ’14)(π‘₯βˆ’14)<br />Β <br />(π‘₯βˆ’14)(π‘₯βˆ’14)=π‘₯2βˆ’14π‘₯βˆ’14π‘₯+196Β <br />Β <br />(π‘₯βˆ’14)2<br />Β <br />
  12. 12. Difference of Squares<br />The -56π‘₯2𝑦 cancels out 56π‘₯2𝑦 leaving us with 64π‘₯4βˆ’49𝑦2<br />Β <br />Factor out the two monomials!<br />Use FOIL to double check your work!<br />A difference of Square's is when a square number is subtracted from another square number.<br />Example: 64π‘₯4βˆ’49𝑦2<br />Β <br />(8π‘₯2+7y)(Β 8π‘₯2βˆ’7y)<br />Β <br />(8π‘₯2+7y)Β 8π‘₯2βˆ’7y=64π‘₯4βˆ’56π‘₯2𝑦+56π‘₯2π‘¦βˆ’49𝑦2<br />Β <br />(8π‘₯2+7y)(Β 8π‘₯2βˆ’7y)<br />Β <br />