Find the quadratic polynomial, sum of whose zeroes is 0 and their prod

1.	Find the quadratic polynomial, sum of whose zeroes is 0 and their product is -1. Hence, find the zeroes of the polynomial.
Answer» Let α and β be the zeroes of the required polynomial f(x). Then (α + β) = 0 and αβ = -1 ∴f(x) = x² ˗ (α + β)x + αβ ⇒ f(x) = x² ˗ 0x + (-1) ⇒ f(x) = x² ˗ 1 Hence, required polynomial f(x) = x² ˗ 1. ∴f(x) = 0 ⇒ x² ˗ 1 = 0 ⇒ (x + 1) (x – 1) = 0 ⇒ (x + 1) = 0 or (x – 1) = 0 ⇒ x = -1 or x = 1 So, the zeroes of f(x) are -1 and 1.

Find the quadratic polynomial, sum of whose zeroes is 0 and their product is -1. Hence, find the zeroes of the polynomial.

Answer»

Let α and β be the zeroes of the required polynomial f(x).

Then (α + β) = 0 and αβ = -1

∴f(x) = x² ˗ (α + β)x + αβ

⇒ f(x) = x² ˗ 0x + (-1)

⇒ f(x) = x² ˗ 1

Hence,

required polynomial f(x) = x² ˗ 1.

∴f(x) = 0 ⇒ x² ˗ 1 = 0

⇒ (x + 1) (x – 1) = 0

⇒ (x + 1) = 0 or (x – 1) = 0

⇒ x = -1 or x = 1

So, the zeroes of f(x) are -1 and 1.