Are the following statements ‘true’ or ‘False’? Justify your answer.
If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.
If the zeroes of a quadratic polynomial ax2 + 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) = ax2 + bx + c
Sum of the zeroes = - (coefficient of x) ÷ coefficient of x2
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 x2
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.