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If there is a multiple root of order 3 for the equation x4 − 2 x3 + 2 x − a = 0 , then the other roots is
- a)-1
- b)0
- c)1
- d)2
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
If there is a multiple root of order 3 for the equation x4 − 2 x...
Explanation:
Multiple Root of Order 3:
- A multiple root of order 3 means that a particular root is a solution to the polynomial equation and its first two derivatives are also equal to zero at that root. In this case, the root occurs three times in the factorization of the polynomial.
Given Equation:
- The equation provided is x^4 - 2x^3 + 2x - a = 0
Other Roots:
- Since one of the roots is a multiple root of order 3, the other roots must be distinct.
- The sum of the roots of a polynomial is equal to the negative of the coefficient of x^(n-1) divided by the coefficient of x^n. In this case, the sum of the roots is 2/1 = 2.
- If one root is repeated thrice, the sum of the roots will be 3 times the repeated root. Therefore, the repeated root is -1, and the sum of all roots is -3.
- Since the sum of all roots is 2, the other distinct root must be 2 - (-3) = 5.
- Therefore, the other roots are -1, 5.
Conclusion:
- The correct answer is option A, which states that the other root is -1.
Multiple Root of Order 3:
- A multiple root of order 3 means that a particular root is a solution to the polynomial equation and its first two derivatives are also equal to zero at that root. In this case, the root occurs three times in the factorization of the polynomial.
Given Equation:
- The equation provided is x^4 - 2x^3 + 2x - a = 0
Other Roots:
- Since one of the roots is a multiple root of order 3, the other roots must be distinct.
- The sum of the roots of a polynomial is equal to the negative of the coefficient of x^(n-1) divided by the coefficient of x^n. In this case, the sum of the roots is 2/1 = 2.
- If one root is repeated thrice, the sum of the roots will be 3 times the repeated root. Therefore, the repeated root is -1, and the sum of all roots is -3.
- Since the sum of all roots is 2, the other distinct root must be 2 - (-3) = 5.
- Therefore, the other roots are -1, 5.
Conclusion:
- The correct answer is option A, which states that the other root is -1.
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If there is a multiple root of order 3 for the equation x4 − 2 x3+ 2 x − a = 0 , then the other roots isa)-1b)0c)1d)2Correct answer is option 'A'. Can you explain this answer?
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