MP Board · Class 10 · Mathematics · PolynomialsCan a quadratic polynomial have no zeroes? Explain with a graphical reference.

Yes, a quadratic polynomial can have no real zeroes. Geometrically, the graph of a quadratic polynomial $y = ax^2 + bx + c$ is a parabola that either opens upwards or downwards. If the parabola lies entirely above the x-axis or entirely below the x-axis, it never intersects or touches the x-axis at any point. In terms of the discriminant ($D = b^2 - 4ac$), this occurs when $D < 0$. Since there are no intersection points with the x-axis, the quadratic polynomial possesses no real zeroes whatsoever.