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The two roots of the cubic equation are complex then the third root is?
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The two roots of the cubic equation are complex then the third root is...
When considering the formula for the solution to a degree three polynomial equation, I observe that it involves two different square roots inside of two different third roots. From the standpoint of complex analysis, each square root represents two different possible branches, and each third root represents three different possible branches. So in principle the cubic formula appears to give possibly 3⋅2⋅3⋅2=36 different roots. Of course we know that a degree three polynomial equation really has only three solutions.
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The two roots of the cubic equation are complex then the third root is...
Explanation:
Given: The two roots of the cubic equation are complex.
Explanation of complex roots:
- Complex roots always occur in conjugate pairs for polynomial equations with real coefficients.
- If two roots are complex, the third root must also be complex and the complex conjugate of the second root.
Sum of roots:
- The sum of the roots of a cubic equation is given by: -b/a, where a is the coefficient of x^3 and b is the coefficient of x^2.
- Since the two roots are complex, the sum of the roots will still be a real number.
Product of roots:
- The product of the roots of a cubic equation is given by: -d/a, where d is the constant term.
- Since the product of the two complex roots is also real, the third root must be complex to maintain the product as real.
Conclusion:
- In conclusion, if two roots of a cubic equation are complex, the third root will also be complex to maintain the properties of the polynomial equation and ensure that the sum and product of the roots remain real numbers.
Given: The two roots of the cubic equation are complex.
Explanation of complex roots:
- Complex roots always occur in conjugate pairs for polynomial equations with real coefficients.
- If two roots are complex, the third root must also be complex and the complex conjugate of the second root.
Sum of roots:
- The sum of the roots of a cubic equation is given by: -b/a, where a is the coefficient of x^3 and b is the coefficient of x^2.
- Since the two roots are complex, the sum of the roots will still be a real number.
Product of roots:
- The product of the roots of a cubic equation is given by: -d/a, where d is the constant term.
- Since the product of the two complex roots is also real, the third root must be complex to maintain the product as real.
Conclusion:
- In conclusion, if two roots of a cubic equation are complex, the third root will also be complex to maintain the properties of the polynomial equation and ensure that the sum and product of the roots remain real numbers.
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The two roots of the cubic equation are complex then the third root is?
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The two roots of the cubic equation are complex then the third root is? for CA Foundation 2025 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The two roots of the cubic equation are complex then the third root is? covers all topics & solutions for CA Foundation 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The two roots of the cubic equation are complex then the third root is?.
The two roots of the cubic equation are complex then the third root is? for CA Foundation 2025 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The two roots of the cubic equation are complex then the third root is? covers all topics & solutions for CA Foundation 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The two roots of the cubic equation are complex then the third root is?.
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