SOLUTION: Find the zeros of {{{x^3-x^2-2x+2}}} and the multiplicity of each.

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Question 149763: Find the zeros of x%5E3-x%5E2-2x%2B2 and the multiplicity of each.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, let's list all of the possible rational zeros.

Any rational zero can be found through this formula

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 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




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Now let's test each possible rational root with use of synthetic division




Let's see if the possible zero 1 is really a root for the function x%5E3-x%5E2-2x%2B2


So let's make the synthetic division table for the function x%5E3-x%5E2-2x%2B2 given the possible zero 1:
1|1-1-22
| 10-2
10-20

Since the remainder 0 (the right most entry in the last row) is equal to zero, this means that 1 is a zero of x%5E3-x%5E2-2x%2B2


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Let's see if the possible zero 2 is really a root for the function x%5E3-x%5E2-2x%2B2


So let's make the synthetic division table for the function x%5E3-x%5E2-2x%2B2 given the possible zero 2:
2|1-1-22
| 220
1102

Since the remainder 2 (the right most entry in the last row) is not equal to zero, this means that 2 is not a zero of x%5E3-x%5E2-2x%2B2


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Let's see if the possible zero -1 is really a root for the function x%5E3-x%5E2-2x%2B2


So let's make the synthetic division table for the function x%5E3-x%5E2-2x%2B2 given the possible zero -1:
-1|1-1-22
| -120
1-202

Since the remainder 2 (the right most entry in the last row) is not equal to zero, this means that -1 is not a zero of x%5E3-x%5E2-2x%2B2


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Let's see if the possible zero -2 is really a root for the function x%5E3-x%5E2-2x%2B2


So let's make the synthetic division table for the function x%5E3-x%5E2-2x%2B2 given the possible zero -2:
-2|1-1-22
| -26-8
1-34-6

Since the remainder -6 (the right most entry in the last row) is not equal to zero, this means that -2 is not a zero of x%5E3-x%5E2-2x%2B2



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So to recap, we only found one rational zero. So the only rational zero for the function x%5E3-x%5E2-2x%2B2 is 1

Now if we go back to the corresponding synthetic division table for the test zero 1, we get

1|1-1-22
| 10-2
10-20


Now looking at the bottom row of coefficients, we see the first three numbers: 1, 0 and -2

These coefficients form the quotient x%5E2%2B0x-2 or just x%5E2-2

This means that %28x%5E3-x%5E2-2x%2B2%29%2F%28x-1%29=x%5E2-2 or x%5E3-x%5E2-2x%2B2=%28x-1%29%28x%5E2-2%29

x%5E2-2=0 Set the quotient equal to zero


x%5E2=2 Add 2 to both sides.


x=0%2B-sqrt%282%29 Take the square root of both sides.


x=sqrt%282%29 or x=-sqrt%282%29 Break up the expression.


So the remaining two zeros are x=sqrt%282%29 or x=-sqrt%282%29


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Answer:

So the zeros of x%5E3-x%5E2-2x%2B2 are:

1, x=sqrt%282%29 or x=-sqrt%282%29 where each zero has a multiplicity of 1.