SOLUTION: FACTOR 4X^2-12X+5 3X^2-5X-2 3Y^2+7Y-6

Question 117667: FACTOR

4X^2-12X+5
3X^2-5X-2
3Y^2+7Y-6
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
#1

Looking at we can see that the first term is and the last term is where the coefficients are 4 and 5 respectively.

Now multiply the first coefficient 4 and the last coefficient 5 to get 20. Now what two numbers multiply to 20 and add to the middle coefficient -12? Let's list all of the factors of 20:

Factors of 20:
1,2,4,5,10,20

-1,-2,-4,-5,-10,-20 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 20
1*20
2*10
4*5
(-1)*(-20)
(-2)*(-10)
(-4)*(-5)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12

First Number	Second Number	Sum
1	20	1+20=21
2	10	2+10=12
4	5	4+5=9
-1	-20	-1+(-20)=-21
-2	-10	-2+(-10)=-12
-4	-5	-4+(-5)=-9

From this list we can see that -2 and -10 add up to -12 and multiply to 20

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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Answer:

So factors to

#2

Looking at we can see that the first term is and the last term is where the coefficients are 3 and -2 respectively.

Now multiply the first coefficient 3 and the last coefficient -2 to get -6. Now what two numbers multiply to -6 and add to the middle coefficient -5? Let's list all of the factors of -6:

Factors of -6:
1,2,3,6

-1,-2,-3,-6 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -6
(1)*(-6)
(2)*(-3)
(-1)*(6)
(-2)*(3)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5

First Number	Second Number	Sum
1	-6	1+(-6)=-5
2	-3	2+(-3)=-1
-1	6	-1+6=5
-2	3	-2+3=1

From this list we can see that 1 and -6 add up to -5 and multiply to -6

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

#3

Looking at we can see that the first term is and the last term is where the coefficients are 3 and -6 respectively.

Now multiply the first coefficient 3 and the last coefficient -6 to get -18. Now what two numbers multiply to -18 and add to the middle coefficient 7? Let's list all of the factors of -18:

Factors of -18:
1,2,3,6,9,18

-1,-2,-3,-6,-9,-18 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -18
(1)*(-18)
(2)*(-9)
(3)*(-6)
(-1)*(18)
(-2)*(9)
(-3)*(6)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7

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 this list we can see that -2 and 9 add up to 7 and multiply to -18

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

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