SOLUTION: rewrite middle term as the sum of two terms and factor completely
1.12w^2+19w+4
2.4z^3-18z^2-10z
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Question 87196: rewrite middle term as the sum of two terms and factor completely
1.12w^2+19w+4
2.4z^3-18z^2-10z
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
In order to factor , first multiply 12 and 4 to get 48 and we need to ask ourselves: What two numbers multiply to 48 and add to 19? Lets find out by listing all of the possible factors of 48
Factors:
1,2,3,4,6,8,12,16,24,48,
-1,-2,-3,-4,-6,-8,-12,-16,-24,-48, List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to 48.
1*48=48
2*24=48
3*16=48
4*12=48
6*8=48
(-1)*(-48)=48
(-2)*(-24)=48
(-3)*(-16)=48
(-4)*(-12)=48
(-6)*(-8)=48
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 19? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 19
| First Number | | | Second Number | | | Sum |
| 1 | | | 48 | || | 1+48=49 |
| 2 | | | 24 | || | 2+24=26 |
| 3 | | | 16 | || | 3+16=19 |
| 4 | | | 12 | || | 4+12=16 |
| 6 | | | 8 | || | 6+8=14 |
| -1 | | | -48 | || | -1+(-48)=-49 |
| -2 | | | -24 | || | -2+(-24)=-26 |
| -3 | | | -16 | || | -3+(-16)=-19 |
| -4 | | | -12 | || | -4+(-12)=-16 |
| -6 | | | -8 | || | -6+(-8)=-14 |
We can see from the table that 3 and 16 add to 19. So the two numbers that multiply to 48 and add to 19 are: 3 and 16
So the original quadratic
breaks down to this (just replace with the two numbers that multiply to 48 and add to 19, which are: 3 and 16)
Group the first two terms together and the last two terms together like this:
Factor a 3 out of the first group and factor a 4 out of the second group.
Now since we have a common term we can combine the two terms. Notice if we let we would get . Since we have that common term , we are able to combine and
Combine like terms.
Answer:
So the quadratic factors to
Notice how foils back to our original problem . This verifies our answer.
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Factor out a "z"
Now lets factor the quadratic inside the parenthesis
In order to factor , first multiply 4 and -10 to get -40 and we need to ask ourselves: What two numbers multiply to -40 and add to -18? Lets find out by listing all of the possible factors of -40
Factors:
1,2,4,5,8,10,20,40,
-1,-2,-4,-5,-8,-10,-20,-40, List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to -40.
(-1)*(40)=-40
(-2)*(20)=-40
(-4)*(10)=-40
(-5)*(8)=-40
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 | | | -40 | || | 1+(-40)=-39 |
| 2 | | | -20 | || | 2+(-20)=-18 |
| 4 | | | -10 | || | 4+(-10)=-6 |
| 5 | | | -8 | || | 5+(-8)=-3 |
| -1 | | | 40 | || | (-1)+40=39 |
| -2 | | | 20 | || | (-2)+20=18 |
| -4 | | | 10 | || | (-4)+10=6 |
| -5 | | | 8 | || | (-5)+8=3 |