Question 123010: I neeed help asap please.
Use Synthetic division to find the 3 zeros of 
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Use Synthetic division to find the 3 zeros of
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-5)....1....7....2....-40
........1....2....-8...|..0
-4)......1....-2...|..0
2)........1....|..0
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The zeroes are x=-5, x=-4, x=2
Cheers,
Stan H.
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
Any rational zero can be found through this equation
 where p and q are the factors of the last and first coefficients
So let's list the factors of -40 (the last coefficient):
Now let's list the factors of 1 (the first coefficient):
Now let's divide each factor of the last coefficient by each factor of the first coefficient
Now simplify
These are all the distinct rational zeros of the function that could occur
To save time, I'm only going to use synthetic division on the possible zeros that are actually zeros of the function.
Otherwise, I would have to use synthetic division on every possible root (there are 16 possible roots, so that means there would be at most 16 synthetic division tables).
However, you might be required to follow this procedure, so this is why I'm showing you how to set up a problem like this
When you graph this polynomial, you will see that  is a zero. So we'll use this for the synthetic division
Now set up the synthetic division table by placing the zero in the upper left corner and placing the coefficients of the polynomial to the right of the test zero.
Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
Multiply 2 by 1 and place the product (which is 2) right underneath the second coefficient (which is 7)
Add 2 and 7 to get 9. Place the sum right underneath 2.
Multiply 2 by 9 and place the product (which is 18) right underneath the third coefficient (which is 2)
Add 18 and 2 to get 20. Place the sum right underneath 18.
Multiply 2 by 20 and place the product (which is 40) right underneath the fourth coefficient (which is -40)
Add 40 and -40 to get 0. Place the sum right underneath 40.
Since the last column adds to zero, we have a remainder of zero. This means  is a factor of 
Now lets look at the bottom row of coefficients:
The first 3 coefficients (1,9,20) form the quotient
So 
You can use this online polynomial division calculator to check your work
Basically factors to 
Now lets break down further
Looking at  we can see that the first term is  and the last term is  where the coefficients are 1 and 20 respectively.
Now multiply the first coefficient 1 and the last coefficient 20 to get 20. Now what two numbers multiply to 20 and add to the middle coefficient 9? Let's list all of the factors of 20:
Factors of 20:
1,2,4,5,10,20
-1,-2,-4,-5,-10,-20 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 20
1*20
2*10
4*5
(-1)*(-20)
(-2)*(-10)
(-4)*(-5)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 9
| First Number | Second Number | Sum | | 1 | 20 | 1+20=21 | | 2 | 10 | 2+10=12 | | 4 | 5 | 4+5=9 | | -1 | -20 | -1+(-20)=-21 | | -2 | -10 | -2+(-10)=-12 | | -4 | -5 | -4+(-5)=-9 |
From this list we can see that 4 and 5 add up to 9 and multiply to 20
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
 Now reintroduce the factor
Now set each factor equal to zero:
 ,  or 
Now solve for x for each factor:
,  or 
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Answer:
So the zeros of  are , or
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