SOLUTION: please help with factoring 3x^2y^3-3x^2y x^2+144 5x^2-14x+16 2x^2+11x-9 Thank you.

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.

-------------------------------------------------------

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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