Question

The graph of a quadratic polynomial f(x) passes through (5,0), (0, -1) and (-2, 0). The two factors of the polynomial are

• A. \((x + 2), (x - 5)\)
• B. \((x + 5), (x - 2)\)
• C. \((x + 1), (x - 5)\)
• D. \((x - 1), (x + 2)\)

Hint:

The point \((0, -1)\) is the y-intercept and is useful for finding the constant scaling factor of the polynomial, but the x-intercepts directly provide the linear factors.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
A quadratic polynomial \(f(x)\) has roots corresponding to the x-intercepts of its graph.
The points given are \((5,0)\), \((0,-1)\), and \((-2,0)\).
Among these, the points on the x-axis are \((5,0)\) and \((-2,0)\).

Step 2: Key Formula or Approach:
If a polynomial \(f(x)\) passes through \((\alpha, 0)\), then \(\alpha\) is a zero of the polynomial.
By the Factor Theorem, if \(\alpha\) is a zero of \(f(x)\), then \((x - \alpha)\) is a factor of \(f(x)\).

Step 3: Detailed Explanation:

• Identify the x-intercepts from the given coordinates:
The graph crosses the x-axis at \(x = 5\) and \(x = -2\).

• Apply the Factor Theorem:
Since \(x = 5\) is a zero, \((x - 5)\) must be a factor of the polynomial.
Since \(x = -2\) is a zero, \((x - (-2)) = (x + 2)\) must be a factor of the polynomial.

• Therefore, the two linear factors of the quadratic polynomial are \((x + 2)\) and \((x - 5)\).


Step 4: Final Answer:
The two factors of the polynomial are \((x + 2)\) and \((x - 5)\).