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The graph of opens downward and lies entirely below the -axis. Which statement is correct?

Class 10PolynomialsAlgebra — Polynomials
  • Aa > 0 and the polynomial has no real zeroes
  • Ba < 0 and the polynomial has no real zeroes
  • Ca < 0 and the polynomial has two real zeroes
  • Da < 0 and the polynomial has exactly one real zero

Step-by-Step Solution

Opening downward means . Lying entirely below the -axis means the curve never meets it, so there are no -intercepts and hence no real zeroes . 'Two real zeroes' would need the curve to cross the axis twice, and '' would make it open upward.
Answer: B) a < 0 and the polynomial has no real zeroes

Key idea: Geometric meaning of zeros

Zeros of are exactly the **-coordinates where the graph meets the -axis**.

- **Linear** = ax + : graph is a straight line, crosses -axis ONCE → exactly 1 zero.
- **Quadratic** : graph is a parabola. Possible cases:
- Crosses -axis at 2 points → 2 distinct real zeros.
- Touches -axis at 1 point → 1 repeated zero (vertex on -axis).
- Does not meet -axis → no real zeros.
- **Cubic** + …: shape is a curve that crosses 1, 2, or 3 times → 1, 2, or 3 real zeros.

So 'find the zeros' is the algebraic version of 'find where the graph crosses the -axis'.

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