Question

Find a quadratic polynomial whose zeroes are \((5 - 2\sqrt{3})\) and \((5 + 2\sqrt{3})\).

Hint:

Irrational zeroes always occur in conjugate pairs for polynomials with rational coefficients. If one zero is \(a - \sqrt{b}\), the other must be \(a + \sqrt{b}\).
Sum is always \(2a\) and product is \(a^2 - b\).

Answer 1

Step 1: Understanding the Concept:
A quadratic polynomial with zeroes \(\alpha\) and \(\beta\) can be written as \(k[x^2 - (\alpha + \beta)x + \alpha\beta]\), where \(k\) is a constant.
Step 2: Key Formula or Approach:
Sum of zeroes (\(S\)) = \(\alpha + \beta\)
Product of zeroes (\(P\)) = \(\alpha\beta\)
Polynomial \(p(x) = x^2 - Sx + P\)
Step 3: Detailed Explanation:
Let \(\alpha = 5 - 2\sqrt{3}\) and \(\beta = 5 + 2\sqrt{3}\).
Find the Sum of zeroes (\(S\)):
\[ S = \alpha + \beta = (5 - 2\sqrt{3}) + (5 + 2\sqrt{3}) = 10 \]
Find the Product of zeroes (\(P\)):
\[ P = \alpha\beta = (5 - 2\sqrt{3})(5 + 2\sqrt{3}) \]
Using identity \((a-b)(a+b) = a^2 - b^2\):
\[ P = (5)^2 - (2\sqrt{3})^2 = 25 - (4 \times 3) = 25 - 12 = 13 \]
The quadratic polynomial is:
\[ p(x) = x^2 - (10)x + 13 \]
Step 4: Final Answer:
The required quadratic polynomial is \(x^2 - 10x + 13\).