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Let f(x) = 3x^3- 4x + k. If f(x) is divisible by x - k, find the remainder when f(x)...

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christiano-cr7 | (Level 1) Salutatorian

Posted August 27, 2013 at 1:20 AM via web

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Let f(x) = 3x^3- 4x + k. If f(x) is divisible by x - k, find the remainder when

f(x) is divided by x + k.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted August 27, 2013 at 6:26 PM (Answer #2)

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You need to use reminder theorem, such that:

`f(x) = (x + k)(ax^2 + bx + c) + d`

d represents the reminder

ax^2 + bx + c represents the quotient

Replacing -k for x yields:

`f(-k) = d`

`f(-k) = -3k^3 + 4k + k => -3k^3 + 5k = d`

The problem provides the information that` f(x)` is divisible by `x - k` , hence `f(k) = 0` , such that:

`f(k) = 3k^3- 4k+ k => 3k^3 - 3k = 0 => k(k^2 - 1) = 0`

Using zero product rule, yields:

`k = 0`

`k^2 - 1 = 0 => k = +-1`

Replacing 0 for k yields:

`-3*0^3 + 5*0 = d => d = 0`

Since the problem did not provide the information that the polynomial is divisible by `x + k,` hence `k!=0.`

If `k = 1` yields:

`-3 + 5 = d => d = 2`

If `k = -1` yields:

`3 - 5 = d => d = -2`

Hence, evaluating the reminder when f(x) is divided by `x + k ` yields `d = +-2.`

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