SOLUTION: FACTORING: 16y^2 + 24y + 9 there is no GCF of any of these numbers

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Question 146140This question is from textbook
: FACTORING: 16y^2 + 24y + 9 there is no GCF of any of these numbers
 This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at 16y%5E2%2B24y%2B9 we can see that the first term is 16y%5E2 and the last term is 9 where the coefficients are 16 and 9 respectively.

Now multiply the first coefficient 16 and the last coefficient 9 to get 144. Now what two numbers multiply to 144 and add to the middle coefficient 24? 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 24? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 24

First NumberSecond NumberSum
11441+144=145
2722+72=74
3483+48=51
4364+36=40
6246+24=30
8188+18=26
9169+16=25
121212+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



From this list we can see that 12 and 12 add up to 24 and multiply to 144


Now looking at the expression 16y%5E2%2B24y%2B9, replace 24y with 12y%2B12y (notice 12y%2B12y adds up to 24y. So it is equivalent to 24y)

16y%5E2%2Bhighlight%2812y%2B12y%29%2B9


Now let's factor 16y%5E2%2B12y%2B12y%2B9 by grouping:


%2816y%5E2%2B12y%29%2B%2812y%2B9%29 Group like terms


4y%284y%2B3%29%2B3%284y%2B3%29 Factor out the GCF of 4y out of the first group. Factor out the GCF of 3 out of the second group


%284y%2B3%29%284y%2B3%29 Since we have a common term of 4y%2B3, we can combine like terms

So 16y%5E2%2B12y%2B12y%2B9 factors to %284y%2B3%29%284y%2B3%29


So this also means that 16y%5E2%2B24y%2B9 factors to %284y%2B3%29%284y%2B3%29 (since 16y%5E2%2B24y%2B9 is equivalent to 16y%5E2%2B12y%2B12y%2B9)


note: %284y%2B3%29%284y%2B3%29 is equivalent to %284y%2B3%29%5E2 since the term 4y%2B3 occurs twice. So 16y%5E2%2B24y%2B9 also factors to %284y%2B3%29%5E2



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Answer:
So 16y%5E2%2B24y%2B9 factors to %284y%2B3%29%5E2