SOLUTION: I need help with this factoring problem if you could????? 4s^3+s^6-45= Thanks for all your help!!!!!

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Question 194980: I need help with this factoring problem if you could?????
4s^3+s^6-45=
Thanks for all your help!!!!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
4s%5E3%2Bs%5E6-45 Start with the given expression.


s%5E6%2B4s%5E3-45 Rearrange the terms.


Looking at s%5E6%2B4s%5E3-45 we can see that the first term is s%5E6 and the last term is -45 where the coefficients are 1 and -45 respectively.

Now multiply the first coefficient 1 and the last coefficient -45 to get -45. Now what two numbers multiply to -45 and add to the middle coefficient 4? Let's list all of the factors of -45:



Factors of -45:
1,3,5,9,15,45

-1,-3,-5,-9,-15,-45 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -45
(1)*(-45)
(3)*(-15)
(5)*(-9)
(-1)*(45)
(-3)*(15)
(-5)*(9)

note: remember, the product of a negative and a positive number is a negative 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
1-451+(-45)=-44
3-153+(-15)=-12
5-95+(-9)=-4
-145-1+45=44
-315-3+15=12
-59-5+9=4



From this list we can see that -5 and 9 add up to 4 and multiply to -45


Now looking at the expression s%5E6%2B4s%5E3-45, replace 4s%5E3 with -5s%5E3%2B9s%5E3 (notice -5s%5E3%2B9s%5E3 adds up to 4s%5E3. So it is equivalent to 4s%5E3)

s%5E6%2Bhighlight%28-5s%5E3%2B9s%5E3%29-45


Now let's factor s%5E6-5s%5E3%2B9s%5E3-45 by grouping:


%28s%5E6-5s%5E3%29%2B%289s%5E3-45%29 Group like terms


s%5E3%28s%5E3-5%29%2B9%28s%5E3-5%29 Factor out the GCF of s%5E3 out of the first group. Factor out the GCF of 9 out of the second group


%28s%5E3%2B9%29%28s%5E3-5%29 Since we have a common term of s%5E3-5, we can combine like terms

So s%5E6-5s%5E3%2B9s%5E3-45 factors to %28s%5E3%2B9%29%28s%5E3-5%29


So this also means that s%5E6%2B4s%5E3-45 factors to %28s%5E3%2B9%29%28s%5E3-5%29 (since s%5E6%2B4s%5E3-45 is equivalent to s%5E6-5s%5E3%2B9s%5E3-45)



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Answer:
So 4s%5E3%2Bs%5E6-45 completely factors to %28s%5E3%2B9%29%28s%5E3-5%29