f(x) = x^5 - 6x^4 + 11x^3 - 2x^2 - 12x + 8   has (x -2) as a factor. Determine which ONE of the following is true about the set of roots for f(x). a)  2 is a single root with 4 more Real and 0...

f(x) = x^5 - 6x^4 + 11x^3 - 2x^2 - 12x + 8   has (x -2) as a factor. Determine which ONE of the following is true about the set of roots for f(x).

a)  2 is a single root with 4 more Real and 0 Complex

b)  2 is a single root with 2 more Real and 2 Complex

c)  2 is single root with no Real and 4 Complex

d)  2 is a double root with 3 more Real and 0 Complex

e)  2 is a double root with 1 more Real and 2 Complex

f)  2 is a triple root with no Real and 2 Complex

g)  2 is a triple root with 2 more Real and 0 Complex

lemjay | High School Teacher | (Level 3) Senior Educator

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`f(x)= x^5 - 6x^4 + 11x^3 - 2x^2 - 12x + 8`

Since (x-2) is a factor of f(x), then x=2 is one of the roots of this function. To determine its multiplicity, let's divide the polynomial by this root using synthetic division.

`2` `|` `1`  `-6`    `11`   `-2`   `-12`     `8`
`2`   `-8`     `6`       `8`   `-8`
_____________________________
`1`  `-4`      `3`      `4`    `-4`     `0`

Since the last number is 0, this proves that 2 is a root of f(x).
Then, divide the quotient by x=2 again.

`2` `|` `1`   `-4`      `3`       `4`    `-4`
`2`   `-4`   `-2`       `4`
__________________________
`1`   `-2`    `-1`       `2`       `0`

Here, the last number is 0. Hence, x=2 is a root of f(x) again.
Then, divide the quotient by x=2 again.

`2` `|` `1`   `-2`     `-1`       `2`
`2`       `0`   `-2`

_____________________
`1`      `0`     `-1`       `0`

The last number is 0 too. So, x=2 is a root of f(x) again.
Also, divide the quotient by the same value of x.

`2` `|`  `1`     `0`     `-1`
`2`        `4`
_____________
`1`     `2`       `3`

Notice that the last number is not zero. This indicates that 2 is no longer a root of the function.  So, we do not have to divide the quotient by x=2 again.

Since x=2 is a root of f(x) three times this indicates that the factor of the function is:

`f(x)= x^5 - 6x^4 + 11x^3 - 2x^2 - 12x + 8`

`f(x)= (x -2)^3(x^2+0x -1)`

`f(x)=(x-2)^2(x^2-1)`

To determine the other roots, factor x^2-1.

`f(x)=(x-2)^3(x-1)(x+1)`

Base on these, the roots of f(x) are `x=2` with multiplicity of 3,

`x=1` , and `x=-1` .

Hence, among the given choices, it is statement (g)  that is true.