Question

Which of the following graphs represents a polynomial with both zeroes being positive?

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

For a quadratic \(ax^2 + bx + c\) with both positive zeroes, the sum of roots (\(-b/a\)) must be positive and the product of roots (\(c/a\)) must also be positive.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The zeroes of a polynomial are the x-intercepts of its graph. If a zero is positive, the graph must cross or touch the x-axis to the right of the y-axis (where \(x>0\)).
Step 2: Key Formula or Approach:
Look for a graph where all intersection points with the horizontal axis are in the first or fourth quadrants (positive x-region).
Step 3: Detailed Explanation:
1. Analyze the axis: The y-axis divides the plane into negative \(x\) (left) and positive \(x\) (right).
2. Analyze intersection points:
- If the curve crosses the x-axis on the left, it has a negative zero.
- If it crosses at the origin, it has a zero at \(x = 0\).
- For both zeroes to be positive, the entire set of x-intercepts must be on the right side of the origin.
Step 4: Final Answer:
The graph representing a polynomial with both positive zeroes is an upward or downward parabola that intersects the x-axis twice to the right of the y-axis.