Question

14. If a and Bare the zeros of the polynomial p(x)=x2 + px +q, find a polynomial whose zeros are
2α/β&2β/α​

Answer 1

Step-by-step explanation:

Let 2 zeroes be a and b of  polynomial  x² + px + q = 0

sum of roots = a + b = -p/1 = -p

products of roots = ab = q/1 = q

(a + b)² = a² + b² + 2ab ⇒  a² + b² = p² - 2q   

(a-b)² = a² + b² -2ab =  p² - 2q -2q = p² - 4q   

now ques asks for new quadratic eq whose roots are (a+b) ²  and  (a-b)²

so sum of new roots are = (a +b)² + (a-b)² = p² + p² -4q = 2p² - 4q

and  product of roots =  (a+b)²(a-b)² =  (p²)² (p²-4q)² = p⁴ (p⁴ +16q² + 8p²q)

hence new quadratic eq gonna be =

   x² - x(sum of roots) + (products of roots)

  x² - x(2p² - 4q) + p⁴(p⁴ + 16q² +8p²q) = 0

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