SOLUTION: 6g^3-24g^2+24g

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Question 183160This question is from textbook
: 6g^3-24g^2+24g This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
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6g%5E3-24g%5E2%2B24g Start with the given expression


6g%28g%5E2-4g%2B4%29 Factor out the GCF 6g


Now let's focus on the inner expression g%5E2-4g%2B4




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Looking at 1g%5E2-4g%2B4 we can see that the first term is 1g%5E2 and the last term is 4 where the coefficients are 1 and 4 respectively.

Now multiply the first coefficient 1 and the last coefficient 4 to get 4. Now what two numbers multiply to 4 and add to the middle coefficient -4? Let's list all of the factors of 4:



Factors of 4:
1,2

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

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

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


Now which of these pairs add to -4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -4

First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4



From this list we can see that -2 and -2 add up to -4 and multiply to 4


Now looking at the expression 1g%5E2-4g%2B4, replace -4g with -2g%2B-2g (notice -2g%2B-2g adds up to -4g. So it is equivalent to -4g)

1g%5E2%2Bhighlight%28-2g%2B-2g%29%2B4


Now let's factor 1g%5E2-2g-2g%2B4 by grouping:


%281g%5E2-2g%29%2B%28-2g%2B4%29 Group like terms


g%28g-2%29-2%28g-2%29 Factor out the GCF of g out of the first group. Factor out the GCF of -2 out of the second group


%28g-2%29%28g-2%29 Since we have a common term of g-2, we can combine like terms

So 1g%5E2-2g-2g%2B4 factors to %28g-2%29%28g-2%29


So this also means that 1g%5E2-4g%2B4 factors to %28g-2%29%28g-2%29 (since 1g%5E2-4g%2B4 is equivalent to 1g%5E2-2g-2g%2B4)


note: %28g-2%29%28g-2%29 is equivalent to %28g-2%29%5E2 since the term g-2 occurs twice. So 1g%5E2-4g%2B4 also factors to %28g-2%29%5E2



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So our expression goes from 6g%28g%5E2-4g%2B4%29 and factors further to 6g%28g-2%29%5E2


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Answer:

So 6g%5E3-24g%5E2%2B24g factors to 6g%28g-2%29%5E2