Question

If α and β are the zeroes of the quadratic polynomial f(x)=x2-3x-2,then find a quadratic polynomial whose zeroes are 1/2α+β and 1/2β+α.

Answer 1

16x² - 9x + 1

Step-by-step-explaination:

Polynomials in form of x^2 - Sx + P, represent S as sum of roots and P as product of roots.

Here, if a and ß are roots:

• a + ß = 3

• aß = - 2

Square on both sides of a + ß:

a² + ß² + 2aß = 9 → a² + ß² = 9 - 2(-2)

a² + ß² = 9 + 4 = 13

Let the required polynomial be x² - px + q, and 1/(2a+ß) & 1/(2ß+a) are roots, so

p = 1/(2a+ß) + 1/(2ß+a)

p = (2ß+a+2a+ß)/(2a+ß)(2ß+a)

p = (3a+3ß)/(4aß+2a²+2ß²+aß)

p = 3(a+ß)/{4(-2)+2(a²+ß²)+(-2)}

p = 3(3)/{-8+2(13)-2}

p = 9/(-8+26-2) = 9/16

Also,

q = 1/(2a+ß) * 1/(2ß+a)

q = 1/(2a+ß)(2ß+a)

q = 1/(4aß+2a²+2ß²+aß)

q = 1/16

Therefore, polynomial is:

x² - (9/16)x + (1/16)

(16x² - 9x +1)/16

Required polynomial is 16x² - 9x + 1