Are the following statements ‘true’ or ‘False’? Justify your answer.

If the zeroes of a quadratic polynomial ax^{2} + bx + c are both positive, then a, b and c all have the same sign.

If the zeroes of a quadratic polynomial ax^{2} + bx + c are both positive, then a, b and c all have the same sign: **False.**

Let α, β be the zeroes of the polynomial p(x) = ax^{2} + bx + c

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

α + β = - b/a > 0 (∵ α > 0, β > 0 ⇒ α + β > 0)

∴ for – b/a > 0, b and a must have opposite signs.

Product of the zeroes = constant term ÷ coefficient of x^{2}

αβ = c/a > 0 (∵ α,β > 0⇒ αβ > 0)

∴ for c/a > 0, c and a must have same signs.

Case 1: when a > 0

⇒ - b > 0 and c > 0

= b < 0 and c > 0

Case 2: when a < 0

⇒ - b < 0 and c < 0

= b > 0 and c < 0

Hence, the coefficients have different signs.

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