The answer is: no. What we have done above in calculating the sum and product of the zeroes is perfectly correct. The zeroes are non-real, but even then, their sum and product will be real. This should not be surprising. For example, if two irrational numbers sum to a rational number, you would not have been surprised.
Similarly, there should be no surprise if two non-real numbers add to a real number. To summarize, the relation between the sum and product of zeroes, and the coefficients of the polynomial, is universally true — it works in all cases, even if the zeroes themselves are non-real. Solution: To solve this problem, you might think that we need to calculate the two zeroes separately, and then square and add them to obtain our answer.
However, the zeroes of this polynomial are non-real, as can be verified easily. Therefore, we take a different approach. We have:. Do not be surprised at this negative answer! This is because the zeroes were non-real, so there is no reason why their squares should add up to a positive number.
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Math Resources. JEE Material. Become a Teacher. Sign In. Sum and Product of Zeroes in a Quadratic. In any quadratic polynomial: The sum of the zeroes is equal to the negative of the coefficient of x by the coefficient of x 2. Their sum is 13, and their product is Related Sections. Variables, Constants and Expressions. The Value of an Expression. Polynomial Expressions. Polynomials in One Variable. Particular Kinds of Polynomials. Degree of a Polynomial. Linear, Quadratic and Cubic Polynomials.
Polynomials of Degree. Multiplication of Polynomials. Zero of a Linear Polynomial. Zeroes of a Quadratic Polynomial. Zeroes of a Cubic Polynomial. Inferring the locations of zeroes indirectly. Calculating Zero of Quadratic Polynomial.
Division of a Polynomial by a Linear Factor. Division of Polynomials in General.
Degree of the Remainder. Division Algorithm — Linear Divisors. The Remainder Theorem. Factors of a Polynomial. Factorization of Quadratic Polynomials. What is an Identity? Squaring a Trinomial. Cubing a Binomial. An Important Identity. The Importance of Coefficients.
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Zeroes of a Cubic. Dividing Two Polynomials. The Division Algorithm for Linear Divisors. The Division Algorithm for General Divisors. Tracking Only the Coefficients. Learn math from the experts and clarify doubts instantly. Instant doubt clearing live one on one. Filled with action and drama, the Zeroes trilogy unites three powerhouse authors for a thrilling high-stakes series.
Visit him at ScottWesterfeld. Margo Lanagan has been publishing stories for children, young adults and adult readers for twenty-five years. She has won numerous awards, including four World Fantasy Awards.
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