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 = ax^{3} + bx^{2} + cx + d, where given α = β = 0

Clearly a≠ 0

Product of all zeroes = - (constant term) ÷ coefficient of x^{3}

αβγ = - d/a

0 = - d/a

⇒ d = 0 (no constant term)

∴ the polynomial is p(x) = ax^{3} + bx^{2} + cx

Sum of the products of two zeroes at a time = coefficient of x ÷ coefficient of x^{3}

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) = ax^{3} + bx^{2}

Sum of the zeroes = - (coefficient of x^{2}) ÷ coefficient of x^{3}

α + β + γ = - b/a

0 + 0 + γ = - b/a

⇒ b = - γ/a (there exists x^{2} term ∵ b≠ 0)

∴ the polynomial is p(x) = ax^{3} - γx^{2}/a

Hence, the polynomial does not have a linear and a constant term.

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