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Let f(x)=x^6+ax^5+bx^4+x^3+bx^2+ax+1 given that one is a root of a f(x)equals to zero and -1 is not the maximum number of distinct real roots that p can have?
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Let f(x)=x^6+ax^5+bx^4+x^3+bx^2+ax+1 given that one is a root of a f(x...
Explanation:
Given Function:
- f(x) = x^6 + ax^5 + bx^4 + x^3 + bx^2 + ax + 1
Roots:
- Given that one root is a
- -1 is not the maximum number of distinct real roots
Analysis:
- Since one root is a, we have a factor of (x - a)
- By the factor theorem, if f(a) = 0, then (x - a) is a factor of f(x)
- Substituting a into f(x) and simplifying, we get f(a) = a^6 + a^2 + 1 + b(a^4 + a) = 0
- This equation can potentially give us more roots depending on the values of a and b
Maximum Number of Roots:
- The given function is a 6th degree polynomial, so it can have at most 6 roots
- Since one root is already given (a), the maximum number of additional roots can be 5
- However, the question specifies that -1 is not the maximum number of distinct real roots
- This means that the function can have more than 5 roots, and potentially all 6 roots could be real and distinct
Conclusion:
- The function f(x) can have at most 6 roots, with one root given as a
- It can have more than 5 real roots, and the total number of roots depends on the values of a and b
Given Function:
- f(x) = x^6 + ax^5 + bx^4 + x^3 + bx^2 + ax + 1
Roots:
- Given that one root is a
- -1 is not the maximum number of distinct real roots
Analysis:
- Since one root is a, we have a factor of (x - a)
- By the factor theorem, if f(a) = 0, then (x - a) is a factor of f(x)
- Substituting a into f(x) and simplifying, we get f(a) = a^6 + a^2 + 1 + b(a^4 + a) = 0
- This equation can potentially give us more roots depending on the values of a and b
Maximum Number of Roots:
- The given function is a 6th degree polynomial, so it can have at most 6 roots
- Since one root is already given (a), the maximum number of additional roots can be 5
- However, the question specifies that -1 is not the maximum number of distinct real roots
- This means that the function can have more than 5 roots, and potentially all 6 roots could be real and distinct
Conclusion:
- The function f(x) can have at most 6 roots, with one root given as a
- It can have more than 5 real roots, and the total number of roots depends on the values of a and b
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Let f(x)=x^6+ax^5+bx^4+x^3+bx^2+ax+1 given that one is a root of a f(x)equals to zero and -1 is not the maximum number of distinct real roots that p can have?
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Let f(x)=x^6+ax^5+bx^4+x^3+bx^2+ax+1 given that one is a root of a f(x)equals to zero and -1 is not the maximum number of distinct real roots that p can have? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f(x)=x^6+ax^5+bx^4+x^3+bx^2+ax+1 given that one is a root of a f(x)equals to zero and -1 is not the maximum number of distinct real roots that p can have? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x)=x^6+ax^5+bx^4+x^3+bx^2+ax+1 given that one is a root of a f(x)equals to zero and -1 is not the maximum number of distinct real roots that p can have?.
Let f(x)=x^6+ax^5+bx^4+x^3+bx^2+ax+1 given that one is a root of a f(x)equals to zero and -1 is not the maximum number of distinct real roots that p can have? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f(x)=x^6+ax^5+bx^4+x^3+bx^2+ax+1 given that one is a root of a f(x)equals to zero and -1 is not the maximum number of distinct real roots that p can have? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x)=x^6+ax^5+bx^4+x^3+bx^2+ax+1 given that one is a root of a f(x)equals to zero and -1 is not the maximum number of distinct real roots that p can have?.
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