Question 173367: please help with factoring
3x^2y^3-3x^2y
x^2+144
5x^2-14x+16
2x^2+11x-9
Thank you.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! # 1
 Start with the given expression
 Factor out the GCF 
 Factor  to get  (by using the difference of squares)
So factors to
# 2
Note: the expression  really looks like 
Looking at we can see that the first term is  and the last term is  where the coefficients are 1 and 144 respectively.
Now multiply the first coefficient 1 and the last coefficient 144 to get 144. Now what two numbers multiply to 144 and add to the middle coefficient 0? Let's list all of the factors of 144:
Factors of 144:
1,2,3,4,6,8,9,12,16,18,24,36,48,72
-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 144
1*144
2*72
3*48
4*36
6*24
8*18
9*16
12*12
(-1)*(-144)
(-2)*(-72)
(-3)*(-48)
(-4)*(-36)
(-6)*(-24)
(-8)*(-18)
(-9)*(-16)
(-12)*(-12)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 0? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 0
| First Number | Second Number | Sum | | 1 | 144 | 1+144=145 | | 2 | 72 | 2+72=74 | | 3 | 48 | 3+48=51 | | 4 | 36 | 4+36=40 | | 6 | 24 | 6+24=30 | | 8 | 18 | 8+18=26 | | 9 | 16 | 9+16=25 | | 12 | 12 | 12+12=24 | | -1 | -144 | -1+(-144)=-145 | | -2 | -72 | -2+(-72)=-74 | | -3 | -48 | -3+(-48)=-51 | | -4 | -36 | -4+(-36)=-40 | | -6 | -24 | -6+(-24)=-30 | | -8 | -18 | -8+(-18)=-26 | | -9 | -16 | -9+(-16)=-25 | | -12 | -12 | -12+(-12)=-24 |
None of these pairs of factors add to 0. So the expression cannot be factored
# 3
Looking at the expression  , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to  (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,2,4,5,8,10,16,20,40,80
-1,-2,-4,-5,-8,-10,-16,-20,-40,-80
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*80
2*40
4*20
5*16
8*10
(-1)*(-80)
(-2)*(-40)
(-4)*(-20)
(-5)*(-16)
(-8)*(-10)
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | 80 | 1+80=81 | | 2 | 40 | 2+40=42 | | 4 | 20 | 4+20=24 | | 5 | 16 | 5+16=21 | | 8 | 10 | 8+10=18 | | -1 | -80 | -1+(-80)=-81 | | -2 | -40 | -2+(-40)=-42 | | -4 | -20 | -4+(-20)=-24 | | -5 | -16 | -5+(-16)=-21 | | -8 | -10 | -8+(-10)=-18 |
From the table, we can see that there are no pairs of numbers which add to .
So cannot be factored.
# 4
Looking at the expression  , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to  (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,2,3,6,9,18
-1,-2,-3,-6,-9,-18
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*(-18)
2*(-9)
3*(-6)
(-1)*(18)
(-2)*(9)
(-3)*(6)
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | -18 | 1+(-18)=-17 | | 2 | -9 | 2+(-9)=-7 | | 3 | -6 | 3+(-6)=-3 | | -1 | 18 | -1+18=17 | | -2 | 9 | -2+9=7 | | -3 | 6 | -3+6=3 |
From the table, we can see that there are no pairs of numbers which add to . So cannot be factored.
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Note: there's always the possibility that you cannot factor an expression. However, most books will only throw them out in small doses. So I would double check your problems to make sure that you copied them down correctly.
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