Question

If α and β are the zeroes of the quadratic polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α – β)2.

​

Answer 1

Solution:

We have,

f(x) = x2 + px + q

Sum of the zeroes = α + β = -p

Product of the zeroes = αβ = q

From the question,

Sum of the zeroes of new polynomial = (α + β)2 + (α – β)2

= (α + β)2 + α2 + β2 – 2αβ

= (α + β)2 + (α + β)2 – 2αβ – 2αβ

= (- p)2 + (- p)2 – 2 × q – 2 × q

= p2 + p2 – 4q

= p2 – 4q

Product of the zeroes of new polynomial = (α + β)2 (α – β)2

= (- p)2((- p)2 - 4q)

= p2 (p2–4q)

So, the quadratic polynomial is,

x2 – (sum of the zeroes)x + (product of the zeroes)

= x2 – (2p2 – 4q)x + p2(p2 – 4q)

Hence, the required quadratic polynomial is f(x) = k(x2 – (2p2 –4q) x + p2(p2 - 4q)).

# BRAINLY CELB

Answer 2

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