SOLUTION: Hi! Find all real zeros of the polynomial. Use the quadratic formula if necessary. P(x) = 4x^3 - 7x^2 - 10x - 2 thanks for the homework help!

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Question 199775: Hi!
Find all real zeros of the polynomial. Use the quadratic formula if necessary.
P(x) = 4x^3 - 7x^2 - 10x - 2
thanks for the homework help!
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
First, let's find the possible rational zeros of P(x):

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 -2 (the last coefficient):



Now let's list the factors of 4 (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



------------------------------------------------------------------------------------------

Now let's test the possible rational zeros to see which ones are actually roots.





Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
1|4-7-10-2
| 4-3-13
4-3-13-15

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
1/2|4-7-10-2
| 2-5/2-25/4
4-5-25/2-33/4

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
1/4|4-7-10-2
| 1-3/2-23/8
4-6-23/2-39/8

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
2|4-7-10-2
| 82-16
41-8-18

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
-1|4-7-10-2
| -411-1
4-111-3

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
-1|4-7-10-2
| -411-1
4-111-3

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
-1/2|4-7-10-2
| -29/211/4
4-9-11/23/4

Since the remainder (the right most entry in the last row) is not equal to zero, this means that is not a zero of


------------------------------------------------------


Let's see if the possible zero is really a root for the function


So let's make the synthetic division table for the function given the possible zero :
-1/4|4-7-10-2
| -122
4-8-80

Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of


So this tells us that


Note: the quotient results from taking a quarter of the first three values in the bottom row.



Now let's solve to find the next two zeros.


Start with the given equation.


Notice we have a quadratic in the form of where , , and


Let's use the quadratic formula to solve for "x":


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


Break up the fraction.


Reduce.


or Break up the expression.


So the next two roots are or


========================================================================

Answer:

So the three real zeros of are:


, or


If you have any questions, email me at jim_thompson5910@hotmail.com
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