Are the following statements ‘true’ or ‘False’? Justify your answer.
If two of the zeroes of cubic polynomials are zero then it does not have linear and constant terms.
If two of the zeroes of cubic polynomials are zero then it does not have linear and constant terms: True
Let α, β and γ be the zeroes of the polynomial p(x0 = ax3 + bx2 + cx + d, where given α = β = 0
Clearly a≠ 0
Product of all zeroes = - (constant term) ÷ coefficient of x3
αβγ = - d/a
0 = - d/a
⇒ d = 0 (no constant term)
∴ the polynomial is p(x) = ax3 + bx2 + cx
Sum of the products of two zeroes at a time = coefficient of x ÷ coefficient of x3
αβ + βγ + αγ = c/a
0 + (0) γ + (0) γ = c/a
0 = c/a
⇒ c = 0 (no linear term)
∴ the polynomial is p(x) = ax3 + bx2
Sum of the zeroes = - (coefficient of x2) ÷ coefficient of x3
α + β + γ = - b/a
0 + 0 + γ = - b/a
⇒ b = - γ/a (there exists x2 term ∵ b≠ 0)
∴ the polynomial is p(x) = ax3 - γx2/a
Hence, the polynomial does not have a linear and a constant term.