Question 141051This question is from textbook Algebra 1
: how do you factor this polynomial x^2+14x+24
This question is from textbook Algebra 1
Found 2 solutions by jim_thompson5910, vleith:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Looking at  we can see that the first term is  and the last term is  where the coefficients are 1 and 24 respectively.
Now multiply the first coefficient 1 and the last coefficient 24 to get 24. Now what two numbers multiply to 24 and add to the middle coefficient 14? Let's list all of the factors of 24:
Factors of 24:
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 24
1*24
2*12
3*8
4*6
(-1)*(-24)
(-2)*(-12)
(-3)*(-8)
(-4)*(-6)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 14
| First Number | Second Number | Sum | | 1 | 24 | 1+24=25 | | 2 | 12 | 2+12=14 | | 3 | 8 | 3+8=11 | | 4 | 6 | 4+6=10 | | -1 | -24 | -1+(-24)=-25 | | -2 | -12 | -2+(-12)=-14 | | -3 | -8 | -3+(-8)=-11 | | -4 | -6 | -4+(-6)=-10 |
From this list we can see that 2 and 12 add up to 14 and multiply to 24
Now looking at the expression , replace  with  (notice adds up to . So it is equivalent to )
Now let's factor  by grouping:
 Group like terms
 Factor out the GCF of  out of the first group. Factor out the GCF of  out of the second group
 Since we have a common term of  , we can combine like terms
So factors to
So this also means that factors to (since is equivalent to )
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Answer:
So factors to
Answer by vleith(2983) (Show Source):
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