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

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Question 117667: FACTOR

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




Looking at 4x%5E2-12x%2B5 we can see that the first term is 4x%5E2 and the last term is 5 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 NumberSecond NumberSum
1201+20=21
2102+10=12
454+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 4x%5E2-12x%2B5, replace -12x with -2x%2B-10x (notice -2x%2B-10x adds up to -12x. So it is equivalent to -12x)

4x%5E2%2Bhighlight%28-2x%2B-10x%29%2B5


Now let's factor 4x%5E2-2x-10x%2B5 by grouping:


%284x%5E2-2x%29%2B%28-10x%2B5%29 Group like terms


2x%282x-1%29-5%282x-1%29 Factor out the GCF of 2x out of the first group. Factor out the GCF of -5 out of the second group


%282x-5%29%282x-1%29 Since we have a common term of 2x-1, we can combine like terms

So 4x%5E2-2x-10x%2B5 factors to %282x-5%29%282x-1%29


So this also means that 4x%5E2-12x%2B5 factors to %282x-5%29%282x-1%29 (since 4x%5E2-12x%2B5 is equivalent to 4x%5E2-2x-10x%2B5)

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Answer:

So 4x%5E2-12x%2B5 factors to %282x-5%29%282x-1%29






#2




Looking at 3x%5E2-5x-2 we can see that the first term is 3x%5E2 and the last term is -2 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 NumberSecond NumberSum
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+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 3x%5E2-5x-2, replace -5x with 1x%2B-6x (notice 1x%2B-6x adds up to -5x. So it is equivalent to -5x)

3x%5E2%2Bhighlight%281x%2B-6x%29%2B-2


Now let's factor 3x%5E2%2B1x-6x-2 by grouping:


%283x%5E2%2B1x%29%2B%28-6x-2%29 Group like terms


x%283x%2B1%29-2%283x%2B1%29 Factor out the GCF of x out of the first group. Factor out the GCF of -2 out of the second group


%28x-2%29%283x%2B1%29 Since we have a common term of 3x%2B1, we can combine like terms

So 3x%5E2%2B1x-6x-2 factors to %28x-2%29%283x%2B1%29


So this also means that 3x%5E2-5x-2 factors to %28x-2%29%283x%2B1%29 (since 3x%5E2-5x-2 is equivalent to 3x%5E2%2B1x-6x-2)

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Answer:

So 3x%5E2-5x-2 factors to %28x-2%29%283x%2B1%29






#3




Looking at 3y%5E2%2B7y-6 we can see that the first term is 3y%5E2 and the last term is -6 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 NumberSecond NumberSum
1-181+(-18)=-17
2-92+(-9)=-7
3-63+(-6)=-3
-118-1+18=17
-29-2+9=7
-36-3+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 3y%5E2%2B7y-6, replace 7y with -2y%2B9y (notice -2y%2B9y adds up to 7y. So it is equivalent to 7y)

3y%5E2%2Bhighlight%28-2y%2B9y%29%2B-6


Now let's factor 3y%5E2-2y%2B9y-6 by grouping:


%283y%5E2-2y%29%2B%289y-6%29 Group like terms


y%283y-2%29%2B3%283y-2%29 Factor out the GCF of y out of the first group. Factor out the GCF of 3 out of the second group


%28y%2B3%29%283y-2%29 Since we have a common term of 3y-2, we can combine like terms

So 3y%5E2-2y%2B9y-6 factors to %28y%2B3%29%283y-2%29


So this also means that 3y%5E2%2B7y-6 factors to %28y%2B3%29%283y-2%29 (since 3y%5E2%2B7y-6 is equivalent to 3y%5E2-2y%2B9y-6)

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Answer:

So 3y%5E2%2B7y-6 factors to %28y%2B3%29%283y-2%29