SOLUTION: Please help me solve this equation! List the rational roots and determine them:{{{2x^3-7x+2=0}}}

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Question 166084: Please help me solve this equation! List the rational roots and determine them:2x%5E3-7x%2B2=0
Answer by jim_thompson5910(35256) About Me  (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 2 (the last coefficient):



Now let's list the factors of 2 (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 all of the possible zeros:



Let's see if the possible zero 1 is really a root for the function 2x%5E3-7x%2B2


So let's make the synthetic division table for the function 2x%5E3-7x%2B2 given the possible zero 1:
1|20-72
| 22-5
22-5-3

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 2x%5E3-7x%2B2


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Let's see if the possible zero 1%2F2 is really a root for the function 2x%5E3-7x%2B2


So let's make the synthetic division table for the function 2x%5E3-7x%2B2 given the possible zero 1%2F2:
1/2|20-72
| 11/2-13/4
21-13/2-5/4

Since the remainder -5%2F4 (the right most entry in the last row) is NOT equal to zero, this means that 1%2F2 is NOT a zero of 2x%5E3-7x%2B2


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Let's see if the possible zero 2 is really a root for the function 2x%5E3-7x%2B2


So let's make the synthetic division table for the function 2x%5E3-7x%2B2 given the possible zero 2:
2|20-72
| 482
2414

Since the remainder 4 (the right most entry in the last row) is NOT equal to zero, this means that 2 is NOT a zero of 2x%5E3-7x%2B2


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Let's see if the possible zero -1 is really a root for the function 2x%5E3-7x%2B2


So let's make the synthetic division table for the function 2x%5E3-7x%2B2 given the possible zero -1:
-1|20-72
| -225
2-2-57

Since the remainder 7 (the right most entry in the last row) is NOT equal to zero, this means that -1 is NOT a zero of 2x%5E3-7x%2B2


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Let's see if the possible zero -1%2F2 is really a root for the function 2x%5E3-7x%2B2


So let's make the synthetic division table for the function 2x%5E3-7x%2B2 given the possible zero -1%2F2:
-1/2|20-72
| -11/213/4
2-1-13/221/4

Since the remainder 21%2F4 (the right most entry in the last row) is NOT equal to zero, this means that -1%2F2 is NOT a zero of 2x%5E3-7x%2B2


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Let's see if the possible zero -2 is really a root for the function 2x%5E3-7x%2B2


So let's make the synthetic division table for the function 2x%5E3-7x%2B2 given the possible zero -2:
-2|20-72
| -48-2
2-410

Since the remainder 0 (the right most entry in the last row) is equal to zero, this means that -2 is a zero of 2x%5E3-7x%2B2

Also, it turns out that the numbers 2,-4, and 1 (the bottom row of numbers) form the coefficients of the quotient. So this means that %282x%5E3-7x%2B2%29%2F%28x%2B2%29=2x%5E2-4x%2B1


2x%5E3-7x%2B2=%28x%2B2%29%282x%5E2-4x%2B1%29 Now multiply both sides by x%2B2


So this means that 2x%5E3-7x%2B2 factors to %28x%2B2%29%282x%5E2-4x%2B1%29


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Now let's solve 2x%5E2-4x%2B1=0


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=2, b=-4, and c=1


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-%28-4%29+%2B-+sqrt%28+%28-4%29%5E2-4%282%29%281%29+%29%29%2F%282%282%29%29 Plug in a=2, b=-4, and c=1


x+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%282%29%281%29+%29%29%2F%282%282%29%29 Negate -4 to get 4.


x+=+%284+%2B-+sqrt%28+16-4%282%29%281%29+%29%29%2F%282%282%29%29 Square -4 to get 16.


x+=+%284+%2B-+sqrt%28+16-8+%29%29%2F%282%282%29%29 Multiply 4%282%29%281%29 to get 8


x+=+%284+%2B-+sqrt%28+8+%29%29%2F%282%282%29%29 Subtract 8 from 16 to get 8


x+=+%284+%2B-+sqrt%28+8+%29%29%2F%284%29 Multiply 2 and 2 to get 4.


x+=+%284+%2B-+2%2Asqrt%282%29%29%2F%284%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%284%2B2%2Asqrt%282%29%29%2F%284%29 or x+=+%284-2%2Asqrt%282%29%29%2F%284%29 Break up the expression.


x+=+%282%2Bsqrt%282%29%29%2F%282%29 or x+=+%282-sqrt%282%29%29%2F%282%29 Reduce


So the last two zeros are x+=+%282%2Bsqrt%282%29%29%2F%282%29 or x+=+%282-sqrt%282%29%29%2F%282%29


which approximate to x=1.707 or x=0.293



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Answer:

So after all of that, we get the three zeros: x=-2, x=%282%2Bsqrt%282%29%29%2F%282%29 or x=%282-sqrt%282%29%29%2F%282%29


Which in decimal form are x=-2, x=1.707, or x=0.293