SOLUTION: The problem is factoring polynomials a^2-10a+24 and 3x^2+18x+27 Thanks!

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Question 336150: The problem is factoring polynomials
a^2-10a+24
and
3x^2+18x+27
Thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first one to get you started. Please only post one problem at a time. Thanks.



Looking at the expression a%5E2-10a%2B24, we can see that the first coefficient is 1, the second coefficient is -10, and the last term is 24.


Now multiply the first coefficient 1 by the last term 24 to get %281%29%2824%29=24.


Now the question is: what two whole numbers multiply to 24 (the previous product) and add to the second coefficient -10?


To find these two numbers, we need to list all of the factors of 24 (the previous product).


Factors of 24:
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to 24.
1*24 = 24
2*12 = 24
3*8 = 24
4*6 = 24
(-1)*(-24) = 24
(-2)*(-12) = 24
(-3)*(-8) = 24
(-4)*(-6) = 24

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -10:


First NumberSecond NumberSum
1241+24=25
2122+12=14
383+8=11
464+6=10
-1-24-1+(-24)=-25
-2-12-2+(-12)=-14
-3-8-3+(-8)=-11
-4-6-4+(-6)=-10



From the table, we can see that the two numbers -4 and -6 add to -10 (the middle coefficient).


So the two numbers -4 and -6 both multiply to 24 and add to -10


Now replace the middle term -10a with -4a-6a. Remember, -4 and -6 add to -10. So this shows us that -4a-6a=-10a.


a%5E2%2Bhighlight%28-4a-6a%29%2B24 Replace the second term -10a with -4a-6a.


%28a%5E2-4a%29%2B%28-6a%2B24%29 Group the terms into two pairs.


a%28a-4%29%2B%28-6a%2B24%29 Factor out the GCF a from the first group.


a%28a-4%29-6%28a-4%29 Factor out 6 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28a-6%29%28a-4%29 Combine like terms. Or factor out the common term a-4


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Answer:


So a%5E2-10a%2B24 factors to %28a-6%29%28a-4%29.


In other words, a%5E2-10a%2B24=%28a-6%29%28a-4%29.


Note: you can check the answer by expanding %28a-6%29%28a-4%29 to get a%5E2-10a%2B24 or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my website.

Jim