The graph of \(y = f(x)\) is shown in the figure for some polynomials \(f(x)\). The number of zeroes for the polynomials is
Show Hint
Always look strictly at the x-axis. Intersections with the y-axis do not count towards the real zeroes of \(y = f(x)\).
Since the curve lies entirely in the upper half-plane (above the x-axis), the number of real roots is instantly 0.
Step 1: Understanding the Question:
This question is from the topic of Polynomials, specifically focusing on the graphical representation and geometrical meaning of the zeroes of a polynomial.
We are given a graph of a polynomial function \(y = f(x)\) and we need to determine the number of real zeroes of this polynomial.
Step 2: Key Formula or Approach:
The number of real zeroes of a polynomial \(y = f(x)\) is geometrically equal to the number of points where the graph of \(y = f(x)\) intersects or touches the x-axis.
Step 3: Detailed Explanation:
Let us observe the given graph carefully.
The graph shows a wave-like curve (a polynomial of higher degree) that fluctuates up and down.
Crucially, the entire curve lies strictly above the horizontal axis (the x-axis).
The curve never comes down to cross the x-axis, nor does it touch the x-axis at any point.
Since there is no point on the graph where \(y = 0\) (which corresponds to the x-axis), there are no real values of \(x\) for which \(f(x) = 0\).
Therefore, the polynomial has no real zeroes.
Thus, the number of zeroes is \(0\).
Step 4: Final Answer:
The number of zeroes for the polynomial is \(0\).
