Determine the product of all values of k for which the polynomial equation 2x^3-9x^2+12x-k=0 has a double root.

To have a double root the equation would be `2(x-a)^2(x-b)` where a is the double root.

Multiplying this out we get

`2(x^2 - 2ax + a^2)(x-b) = 2x^3 - 4ax^2 + 2a^2x - 2x^2 + 4abx - 2a^2b` `= 2x^3 - 2(2a+b)x^2 + 2(a^2 + 2ab)x + 2(a^2b)`

So we...

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To have a double root the equation would be `2(x-a)^2(x-b)` where a is the double root.

Multiplying this out we get

`2(x^2 - 2ax + a^2)(x-b) = 2x^3 - 4ax^2 + 2a^2x - 2x^2 + 4abx - 2a^2b`
`= 2x^3 - 2(2a+b)x^2 + 2(a^2 + 2ab)x + 2(a^2b)`

So we have
(1) `2(2a+b)=9`
(2) `2a(a+2b)=12`

Solving (1) for b we get
`b=9/2-2a` substituting into (2)
`2a(a+2(9/2-2a)) = 12`
`2a(a+9-4a)=12`
`a(-3a+9) = 6`
`a(a-3) = -2`
`a^2 - 3a + 2 = 0`
`(a-2)(a-1) = 0 so a = 2 or a = 1`
b = `9/2 - 2(2) = 1/2 or b = 9/2 - 2(1) = 5/2`

(1) `2(2(2)+(1/2)) = 2(4+1/2) = 2(9/2) = 9`
(2) `2(2)(2+2(1/2))=4(3) = 12`

(1) `2(2(1)+(5/2))=2(2+5/2)=2(9/2) = 9`
(2) `2(1)(1+2(5/2))=-2(6)=12`

So `k = 2a^2(b) = 2(2)^2(1/2) = 4` or
k = `2(1^2)(5/2) = 5`

So the answer is k = 4 or 5

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