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!

Algebra ->  Polynomials-and-rational-expressions -> 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!      Log On


   

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) About Me  (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



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





Let's see if the possible zero 1 is really a root for the function 4x%5E3-7x%5E2-10x-2


So let's make the synthetic division table for the function 4x%5E3-7x%5E2-10x-2 given the possible zero 1:
1|4-7-10-2
| 4-3-13
4-3-13-15

Since the remainder -15 (the right most entry in the last row) is not equal to zero, this means that 1 is not a zero of 4x%5E3-7x%5E2-10x-2


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Let's see if the possible zero 1%2F2 is really a root for the function 4x%5E3-7x%5E2-10x-2


So let's make the synthetic division table for the function 4x%5E3-7x%5E2-10x-2 given the possible zero 1%2F2:
1/2|4-7-10-2
| 2-5/2-25/4
4-5-25/2-33/4

Since the remainder -33%2F4 (the right most entry in the last row) is not equal to zero, this means that 1%2F2 is not a zero of 4x%5E3-7x%5E2-10x-2


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Let's see if the possible zero 1%2F4 is really a root for the function 4x%5E3-7x%5E2-10x-2


So let's make the synthetic division table for the function 4x%5E3-7x%5E2-10x-2 given the possible zero 1%2F4:
1/4|4-7-10-2
| 1-3/2-23/8
4-6-23/2-39/8

Since the remainder -39%2F8 (the right most entry in the last row) is not equal to zero, this means that 1%2F4 is not a zero of 4x%5E3-7x%5E2-10x-2


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Let's see if the possible zero 2 is really a root for the function 4x%5E3-7x%5E2-10x-2


So let's make the synthetic division table for the function 4x%5E3-7x%5E2-10x-2 given the possible zero 2:
2|4-7-10-2
| 82-16
41-8-18

Since the remainder -18 (the right most entry in the last row) is not equal to zero, this means that 2 is not a zero of 4x%5E3-7x%5E2-10x-2


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Let's see if the possible zero -1 is really a root for the function 4x%5E3-7x%5E2-10x-2


So let's make the synthetic division table for the function 4x%5E3-7x%5E2-10x-2 given the possible zero -1:
-1|4-7-10-2
| -411-1
4-111-3

Since the remainder -3 (the right most entry in the last row) is not equal to zero, this means that -1 is not a zero of 4x%5E3-7x%5E2-10x-2


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Let's see if the possible zero -1 is really a root for the function 4x%5E3-7x%5E2-10x-2


So let's make the synthetic division table for the function 4x%5E3-7x%5E2-10x-2 given the possible zero -1:
-1|4-7-10-2
| -411-1
4-111-3

Since the remainder -3 (the right most entry in the last row) is not equal to zero, this means that -1 is not a zero of 4x%5E3-7x%5E2-10x-2


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Let's see if the possible zero -1%2F2 is really a root for the function 4x%5E3-7x%5E2-10x-2


So let's make the synthetic division table for the function 4x%5E3-7x%5E2-10x-2 given the possible zero -1%2F2:
-1/2|4-7-10-2
| -29/211/4
4-9-11/23/4

Since the remainder 3%2F4 (the right most entry in the last row) is not equal to zero, this means that -1%2F2 is not a zero of 4x%5E3-7x%5E2-10x-2


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Let's see if the possible zero -1%2F4 is really a root for the function 4x%5E3-7x%5E2-10x-2


So let's make the synthetic division table for the function 4x%5E3-7x%5E2-10x-2 given the possible zero -1%2F4:
-1/4|4-7-10-2
| -122
4-8-80

Since the remainder 0 (the right most entry in the last row) is equal to zero, this means that -1%2F4 is a zero of 4x%5E3-7x%5E2-10x-2


So this tells us that 4x%5E3-7x%5E2-10x-2=%284x%2B1%29%28x%5E2-2x-2%29


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



Now let's solve x%5E2-2x-2=0 to find the next two zeros.


x%5E2-2x-2=0 Start with the given equation.


Notice we have a quadratic in the form of Ax%5E2%2BBx%2BC where A=1, B=-2, and C=-2


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


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-2%29+%29%29%2F%282%281%29%29 Plug in A=1, B=-2, and C=-2


x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-2%29+%29%29%2F%282%281%29%29 Negate -2 to get 2.


x+=+%282+%2B-+sqrt%28+4-4%281%29%28-2%29+%29%29%2F%282%281%29%29 Square -2 to get 4.


x+=+%282+%2B-+sqrt%28+4--8+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-2%29 to get -8


x+=+%282+%2B-+sqrt%28+4%2B8+%29%29%2F%282%281%29%29 Rewrite sqrt%284--8%29 as sqrt%284%2B8%29


x+=+%282+%2B-+sqrt%28+12+%29%29%2F%282%281%29%29 Add 4 to 8 to get 12


x+=+%282+%2B-+sqrt%28+12+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%282+%2B-+2%2Asqrt%283%29%29%2F%282%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%282%29%2F%282%29+%2B-+%282%2Asqrt%283%29%29%2F%282%29 Break up the fraction.


x+=+1+%2B-+sqrt%283%29 Reduce.


x+=+1%2Bsqrt%283%29 or x+=+1-sqrt%283%29 Break up the expression.


So the next two roots are x+=+1%2Bsqrt%283%29 or x+=+1-sqrt%283%29


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Answer:

So the three real zeros of P%28x%29+=+4x%5E3+-+7x%5E2+-+10x+-+2 are:


x=-1%2F4, x+=+1%2Bsqrt%283%29 or x+=+1-sqrt%283%29


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