Multiple choice

Find the quadratic polynomial where sum and product of the zeros are $ a$ and $\dfrac{1}{a}$.

  1. $ax^2+a^2x+1=0$
  2. $ax^2-ax+1=0$
  3. $ax^2-a^2x-1=0$
  4. $ax^2-a^2x+1=0$
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D Correct answer
Explanation

A quadratic polynomial is given by x^2 - (sum of zeros)x + (product of zeros). Here, x^2 - ax + 1/a = 0. Multiplying by a gives ax^2 - a^2x + 1 = 0.

AI explanation

The standard form of a quadratic polynomial in terms of its sum and product of zeros is given by the formula k(x^2 - (Sum of zeros)x + (Product of zeros)), where k is any non-zero constant. Substituting the given sum of zeros as 'a' and the product of zeros as '1/a', the polynomial becomes k(x^2 - ax + 1/a). To eliminate the fraction, multiply the entire polynomial by 'a' to get a general polynomial of ka(x^2 - ax + 1/a). Assuming k = 1, multiplying through by 'a' gives ax^2 - a^2x + 1 = 0.

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