SOLUTION: Hello out there,I need some help with some problems,Factor.Write prime if the expression cannot be factored:3a^2+9a+6.The next one is,Factor completely:8g^2j-12g^3hj.The next one i

Algebra ->  Expressions -> SOLUTION: Hello out there,I need some help with some problems,Factor.Write prime if the expression cannot be factored:3a^2+9a+6.The next one is,Factor completely:8g^2j-12g^3hj.The next one i      Log On


   

Question 91399: Hello out there,I need some help with some problems,Factor.Write prime if the expression cannot be factored:3a^2+9a+6.The next one is,Factor completely:8g^2j-12g^3hj.The next one is,Factor.Write prime if the expression cannot be factored:6f^2+5f+2.And the last one is,Factor.Write prime if the expression cannot be factored:12w^2+10w-8.Thanks for any help I get.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


3%2Aa%5E2%2B9%2Aa%2B6 Start with the given expression.



3%28a%5E2%2B3a%2B2%29 Factor out the GCF 3.



Now let's try to factor the inner expression a%5E2%2B3a%2B2



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Looking at the expression a%5E2%2B3a%2B2, we can see that the first coefficient is 1, the second coefficient is 3, and the last term is 2.



Now multiply the first coefficient 1 by the last term 2 to get %281%29%282%29=2.



Now the question is: what two whole numbers multiply to 2 (the previous product) and add to the second coefficient 3?



To find these two numbers, we need to list all of the factors of 2 (the previous product).



Factors of 2:

1,2

-1,-2



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to 2.

1*2 = 2
(-1)*(-2) = 2


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 3:



First NumberSecond NumberSum
121+2=3
-1-2-1+(-2)=-3




From the table, we can see that the two numbers 1 and 2 add to 3 (the middle coefficient).



So the two numbers 1 and 2 both multiply to 2 and add to 3



Now replace the middle term 3a with a%2B2a. Remember, 1 and 2 add to 3. So this shows us that a%2B2a=3a.



a%5E2%2Bhighlight%28a%2B2a%29%2B2 Replace the second term 3a with a%2B2a.



%28a%5E2%2Ba%29%2B%282a%2B2%29 Group the terms into two pairs.



a%28a%2B1%29%2B%282a%2B2%29 Factor out the GCF a from the first group.



a%28a%2B1%29%2B2%28a%2B1%29 Factor out 2 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%28a%2B2%29%28a%2B1%29 Combine like terms. Or factor out the common term a%2B1



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So 3%28a%5E2%2B3a%2B2%29 then factors further to 3%28a%2B2%29%28a%2B1%29



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Answer:



So 3%2Aa%5E2%2B9%2Aa%2B6 completely factors to 3%28a%2B2%29%28a%2B1%29.



In other words, 3%2Aa%5E2%2B9%2Aa%2B6=3%28a%2B2%29%28a%2B1%29.



Note: you can check the answer by expanding 3%28a%2B2%29%28a%2B1%29 to get 3%2Aa%5E2%2B9%2Aa%2B6 or by graphing the original expression and the answer (the two graphs should be identical).




8g%5E2j-12g%5E3hj

4g%5E2j%282-3gh%29 Factor out the GCF 4g%5E2j


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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 6f%5E2%2B5f%2B2, we can see that the first coefficient is 6, the second coefficient is 5, and the last term is 2.



Now multiply the first coefficient 6 by the last term 2 to get %286%29%282%29=12.



Now the question is: what two whole numbers multiply to 12 (the previous product) and add to the second coefficient 5?



To find these two numbers, we need to list all of the factors of 12 (the previous product).



Factors of 12:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to 12.

1*12 = 12
2*6 = 12
3*4 = 12
(-1)*(-12) = 12
(-2)*(-6) = 12
(-3)*(-4) = 12


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:



First NumberSecond NumberSum
1121+12=13
262+6=8
343+4=7
-1-12-1+(-12)=-13
-2-6-2+(-6)=-8
-3-4-3+(-4)=-7




From the table, we can see that there are no pairs of numbers which add to 5. So 6f%5E2%2B5f%2B2 cannot be factored.



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Answer:



So 6%2Af%5E2%2B5%2Af%2B2 doesn't factor at all (over the rational numbers).



So 6%2Af%5E2%2B5%2Af%2B2 is prime.





Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


12%2Aw%5E2%2B10%2Aw-8 Start with the given expression.



2%286w%5E2%2B5w-4%29 Factor out the GCF 2.



Now let's try to factor the inner expression 6w%5E2%2B5w-4



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Looking at the expression 6w%5E2%2B5w-4, we can see that the first coefficient is 6, the second coefficient is 5, and the last term is -4.



Now multiply the first coefficient 6 by the last term -4 to get %286%29%28-4%29=-24.



Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient 5?



To find these two numbers, we need to list all of the factors of -24 (the previous product).



Factors of -24:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to -24.

1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:



First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2




From the table, we can see that the two numbers -3 and 8 add to 5 (the middle coefficient).



So the two numbers -3 and 8 both multiply to -24 and add to 5



Now replace the middle term 5w with -3w%2B8w. Remember, -3 and 8 add to 5. So this shows us that -3w%2B8w=5w.



6w%5E2%2Bhighlight%28-3w%2B8w%29-4 Replace the second term 5w with -3w%2B8w.



%286w%5E2-3w%29%2B%288w-4%29 Group the terms into two pairs.



3w%282w-1%29%2B%288w-4%29 Factor out the GCF 3w from the first group.



3w%282w-1%29%2B4%282w-1%29 Factor out 4 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%283w%2B4%29%282w-1%29 Combine like terms. Or factor out the common term 2w-1



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So 2%286w%5E2%2B5w-4%29 then factors further to 2%283w%2B4%29%282w-1%29



===============================================================



Answer:



So 12%2Aw%5E2%2B10%2Aw-8 completely factors to 2%283w%2B4%29%282w-1%29.



In other words, 12%2Aw%5E2%2B10%2Aw-8=2%283w%2B4%29%282w-1%29.



Note: you can check the answer by expanding 2%283w%2B4%29%282w-1%29 to get 12%2Aw%5E2%2B10%2Aw-8 or by graphing the original expression and the answer (the two graphs should be identical).