SOLUTION: Verify that x = 2 is a root of multiplicity 3 of the equation x^4 - 4x^3 + 16x - 16 = 0. What is the other root?

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Question 141543: Verify that x = 2 is a root of multiplicity 3 of the equation x^4 - 4x^3 + 16x - 16 = 0. What is the other root?
Answer by solver91311(24713) About Me  (Show Source):
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Do the binomial expansion on %28x-2%29%5E3. Using the result as the divisor and x%5E4+-+4x%5E3+%2B+16x+-+16+=+0 as the dividend, perform polynomial long division or synthetic division. Remember to insert 0x%5E2 as a placeholder for the missing 2nd degree term. If x = 2 is a root of multiplicity 3, you will get no remainder and the quotient will be different from x-2.

If the quotient IS x-2, then %28x-2%29 is a root with multiplicity 4.

If there is a remainder, then x-2 is not a root at all.

The quotient, x-a where a%3C%3E2, will be the 4th factor, and you can solve x-a=0 to get the 4th root.