Multiple choice

Find a quadratic polynomial each with the given numbers as the zeroes of the polynomial. $\dfrac {8}{3}, \dfrac {5}{2}$

  1. $6x^2-12x+40$
  2. $6x^2-31x+40$
  3. $6x^2-41x+40$
  4. $6x^2-11x+40$
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B Correct answer
Explanation

For zeroes alpha and beta, the polynomial is k(x^2 - (alpha+beta)x + alpha*beta). Sum = 8/3 + 5/2 = 31/6. Product = 8/3 * 5/2 = 40/6 = 20/3. Polynomial = x^2 - 31/6x + 20/3 = 1/6(6x^2 - 31x + 40).

AI explanation

Using the sum and product of roots to form the polynomial, the sum of the zeroes 8/3 and 5/2 is (16 + 15)/6 = 31/6, and the product is 40/6. A standard quadratic polynomial follows the form x^2 - (sum)x + product, which becomes x^2 - (31/6)x + 40/6. Multiplying the entire expression by 6 to clear the fractions gives 6x^2 - 31x + 40.

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