Multiple choice

If $\alpha$ and $\beta$ are the zeroes of a quadratic polynomial such that $\alpha+\beta=21$ and $\alpha-\beta=5$, then the polynomial is

  1. ${x}^{2}-21x+105$
  2. ${x}^{2}-21x+104$
  3. ${x}^{2}-21x+84$

  4. None of these
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B Correct answer
Explanation

Sum of roots = 21. Difference = 5. (alpha+beta)^2 - (alpha-beta)^2 = 4*alpha*beta. 21^2 - 5^2 = 441 - 25 = 416 = 4*alpha*beta. alpha*beta = 104. Polynomial is x^2 - (sum)x + (product) = x^2 - 21x + 104.

AI explanation

We solve the system of linear equations alpha + beta = 21 and alpha - beta = 5 to find the individual roots. Adding the two equations gives 2 * alpha = 26, so alpha = 13, and substituting this back gives beta = 8. For a quadratic polynomial, the sum of the roots is 21 and the product of the roots is 13 * 8 = 104. Therefore, using the standard form x^2 - (sum)x + (product), the required polynomial is x^2 - 21x + 104.

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