SOLUTION: factoring z^2+18z+45
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Question 119700: factoring z^2+18z+45
Found 2 solutions by scott8148, jim_thompson5910:
Answer by scott8148(6628) (Show Source): You can put this solution on YOUR website!
the b coefficient is a combination of factors of the c term __ (z+15)(z+3)
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 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 18? 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 two negative numbers multiplied together make a positive number
Now which of these pairs add to 18? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 18
| First Number | Second Number | Sum | | 1 | 45 | 1+45=46 |
| 3 | 15 | 3+15=18 |
| 5 | 9 | 5+9=14 |
| -1 | -45 | -1+(-45)=-46 |
| -3 | -15 | -3+(-15)=-18 |
| -5 | -9 | -5+(-9)=-14 |
From this list we can see that 3 and 15 add up to 18 and multiply to 45
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
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