Question

Which of the following quadratic polynomials has zeroes 3 and -10?

• A. \(x^2 + 7x - 30\)
• B. \(x^2 - 7x - 30\)
• C. \(x^2 + 7x + 30\)
• D. \(x^2 - 7x + 30\)

Hint:

Pay close attention to the signs when substituting the sum and product into the formula. The formula has a minus sign before the sum, so `x² - (-7)x` becomes `x² + 7x`.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A quadratic polynomial can be constructed if its zeroes (roots) are known. Let the zeroes be \(\alpha\) and \(\beta\). The polynomial can be expressed in the form \(k(x^2 - (\alpha + \beta)x + \alpha\beta)\), where \(k\) is a non-zero constant.

Step 2: Key Formula or Approach:
1. Find the sum of the zeroes: Sum = \(\alpha + \beta\).
2. Find the product of the zeroes: Product = \(\alpha \beta\).
3. Substitute these values into the formula: \(x^2 - (\text{Sum})x + (\text{Product})\). (Assuming k=1, which is standard for multiple-choice questions).

Step 3: Detailed Explanation:
The given zeroes are \(\alpha = 3\) and \(\beta = -10\).
Calculate the sum of the zeroes:
\[ \text{Sum} = 3 + (-10) = -7 \] Calculate the product of the zeroes:
\[ \text{Product} = 3 \times (-10) = -30 \] Now, construct the polynomial:
\[ x^2 - (\text{Sum})x + (\text{Product}) = x^2 - (-7)x + (-30) \] \[ = x^2 + 7x - 30 \] This matches option (A).

Step 4: Final Answer:
The required quadratic polynomial is \(x^2 + 7x - 30\).