Question

if a and b are zero of polynomial x^2 +px +q, find polynomial whose zero are (a + b) ^2 and (a-b) ^2​

Answer 1

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  

further  u can simplify.  

 

Hope my ans is correct ...........doubts , then enquire.

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