Question

Which of the following quadratic polynomials has zeroes 2 and -2?

• A. \(x^2 + 4\)
• B. \(x^2 - 4\)
• C. \(x^2 - 2x + 4\)
• D. \(x^2 + \sqrt{8}\)

Hint:

Remember the difference of squares identity: \((a-b)(a+b) = a^2 - b^2\). This is very useful for finding a polynomial when the zeroes are opposites of each other (like a and -a).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
If \(\alpha\) and \(\beta\) are the zeroes of a quadratic polynomial, then the polynomial can be written in the form \(k(x - \alpha)(x - \beta)\), where k is a non-zero constant. Another way is to use the sum and product of zeroes: the polynomial is \(k(x^2 - (\text{sum of zeroes})x + (\text{product of zeroes}))\).

Step 2: Key Formula or Approach:
Using the sum and product of zeroes:
Polynomial = \(x^2 - (\alpha + \beta)x + (\alpha \beta)\) (assuming k=1).

Step 3: Detailed Explanation:
We are given the zeroes \(\alpha = 2\) and \(\beta = -2\).
Method 1: Using factors
The factors of the polynomial are \((x - \alpha)\) and \((x - \beta)\).
So the factors are \((x - 2)\) and \((x - (-2)) = (x + 2)\).
The polynomial is the product of these factors:
\[ (x - 2)(x + 2) \] This is in the form of \((a - b)(a + b) = a^2 - b^2\).
\[ = x^2 - 2^2 = x^2 - 4 \] Method 2: Using sum and product of zeroes
Sum of zeroes: \(\alpha + \beta = 2 + (-2) = 0\).
Product of zeroes: \(\alpha \beta = 2 \times (-2) = -4\).
The polynomial is \(x^2 - (\text{sum})x + (\text{product})\).
\[ = x^2 - (0)x + (-4) = x^2 - 4 \]

Step 4: Final Answer:
Both methods give the polynomial \(x^2 - 4\). This matches option (B).