Question

If alpha and beta are zeroes of polynomial x^2-x-2 find a polynimoal whose zereos are alpha^2/beta^2

Answer 1

Answer:

x² - 5x + 4

Step-by-step explanation:

Given polynomial:

x² - x - 2, on comparing the given equation with ax² + bx + c, we get -

• a = 1
• b = - 1
• c = - 2

Sum of zeroes = - b/a

⇒ α + β = - (-1)/1

⇒ α + β = 1

Product of zeroes = c/a

⇒ αβ = - 2/1

⇒ αβ = - 2

Now, we need a polynomial whose zeroes are α² and β².

Sum of zeroes = α² + β²

⇒ α² + β² = (α + β)² - 2αβ

⇒ α² + β² = (1)² - 2(-2)

⇒ α² + β² = 1 + 4

⇒ α² + β² = 5

Product of zeroes = α²β²

⇒ α²β² = (αβ)²

⇒ α²β² = (-2)²

⇒ α²β² = 4

Therefore, the required polynomial is;

= x² - (α² + β²)x + α²β²

= x² - 5x + 4

Hence, the required quadratic polynomial is x² - 5x + 4.