Find the quadratic polynomial, sum of whose zeroes is 8 and their prod

1.	Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.
Answer» Let α and β be the zeroes of the required polynomial f(x). Then (α + β) = 8 and αβ = 12 ∴f(x) = x² ˗ (α + β)x + αβ ⇒ f(x) = x² ˗ 8x + 12 Hence, required polynomial f(x) = x² ˗ 8x + 12 ∴f(x) = 0 ⇒ x² ˗ 8x + 12 = 0 ⇒ x² ˗ (6x + 2x) + 12 = 0 ⇒ x² ˗ 6x ˗ 2x + 12 = 0 ⇒ x (x – 6) – 2 (x – 6) = 0 ⇒ (x – 2) (x – 6) = 0 ⇒ (x – 2) = 0 or (x – 6) = 0 ⇒ x = 2 or x = 6 So, the zeroes of f(x) are 2 and 6.

Find the quadratic polynomial, sum of whose zeroes is 8 and their product is 12. Hence, find the zeroes of the polynomial.

Answer»

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

Then (α + β) = 8 and αβ = 12

∴f(x) = x² ˗ (α + β)x + αβ

⇒ f(x) = x² ˗ 8x + 12

Hence,

required polynomial f(x) = x² ˗ 8x + 12

∴f(x) = 0 ⇒ x² ˗ 8x + 12 = 0

⇒ x² ˗ (6x + 2x) + 12 = 0

⇒ x² ˗ 6x ˗ 2x + 12 = 0

⇒ x (x – 6) – 2 (x – 6) = 0

⇒ (x – 2) (x – 6) = 0

⇒ (x – 2) = 0 or (x – 6) = 0

⇒ x = 2 or x = 6

So, the zeroes of f(x) are 2 and 6.