Find the zeroes of cubic polynomials.i) - x3ii) x2 - x3(iii) x3 –

1.	Find the zeroes of cubic polynomials.i) - x3ii) x2 - x3(iii) x3 – 5x2 + 6x
Answer» i) Given polynomial is y = – x³ f(x) = -x³ ; f(x) = 0 ² x³ = 0 x = \(\sqrt[3]{0}\)= 0 ∴ Zero of the polynomial f(x) is only one i.e., 0. ii) Given that y = x² – x³ f(x) = x² (1 – x) f(x) = 0 ⇒ x² (1 – x) = 0 ⇒ x² = 0 and 1 – x = 0 ⇒ x = 0 and x = 1 ∴ The zeroes of the polynomial f(x) are two i.e., 0 and 1. iii) Given that x³ – 5x² + 6x Let f(x) = x³ – 5x² + 6x = x(x² – 5x + 6) = x(x² – 2x – 3x + 6) = x[x(x – 2) – 3(x – 2)] = x(x – 2) (x – 3) ∴ The zeroes of the polynomial f(x) are x = 0 and x = 2 and x = 3

Find the zeroes of cubic polynomials.i) - x3ii) x2 - x3(iii) x3 – 5x2 + 6x

Answer»

i) Given polynomial is y = – x³

f(x) = -x³ ; f(x) = 0 ²

x³ = 0

x = \(\sqrt[3]{0}\)= 0

∴ Zero of the polynomial f(x) is only one i.e., 0.

ii) Given that y = x² – x³

f(x) = x² (1 – x)

f(x) = 0

⇒ x² (1 – x) = 0

⇒ x² = 0 and 1 – x = 0

⇒ x = 0 and x = 1

∴ The zeroes of the polynomial f(x) are two i.e., 0 and 1.

iii) Given that x³ – 5x² + 6x Let f(x) = x³ – 5x² + 6x

= x(x² – 5x + 6)

= x(x² – 2x – 3x + 6)

= x[x(x – 2) – 3(x – 2)]

= x(x – 2) (x – 3)

∴ The zeroes of the polynomial f(x) are x = 0 and x = 2 and x = 3