Quadratic Polynomial from Given Zeros

Video Explanation

Question

Find a quadratic polynomial whose sum and product of the zeros are:

\[ \text{Sum} = -2\sqrt{3}, \quad \text{Product} = -9 \]

Also, find the zeros of this polynomial by factorisation.

Solution

Step 1: Use the Standard Form

If the sum of zeros is \(S\) and the product is \(P\), then the quadratic polynomial is:

\[ x^2 – Sx + P \]

Step 2: Form the Required Polynomial

Substitute the given values:

\[ x^2 – (-2\sqrt{3})x + (-9) \]

\[ x^2 + 2\sqrt{3}x – 9 \]

Hence, the required quadratic polynomial is:

\[ \boxed{x^2 + 2\sqrt{3}x – 9} \]

Step 3: Factorise the Polynomial

Product of coefficient of \(x^2\) and constant term:

\[ 1 \times (-9) = -9 \]

Split the middle term using \(3\sqrt{3}\) and \(-\sqrt{3}\), since

\[ 3\sqrt{3} – \sqrt{3} = 2\sqrt{3} \quad \text{and} \quad (3\sqrt{3})(-\sqrt{3}) = -9 \]

\[ x^2 + 3\sqrt{3}x – \sqrt{3}x – 9 \]

Grouping the terms:

\[ (x^2 + 3\sqrt{3}x) – (\sqrt{3}x + 9) \]

\[ x(x + 3\sqrt{3}) – \sqrt{3}(x + 3\sqrt{3}) \]

\[ (x – \sqrt{3})(x + 3\sqrt{3}) \]

Step 4: Find the Zeros

\[ (x – \sqrt{3})(x + 3\sqrt{3}) = 0 \]

\[ x – \sqrt{3} = 0 \Rightarrow x = \sqrt{3} \]

\[ x + 3\sqrt{3} = 0 \Rightarrow x = -3\sqrt{3} \]

Conclusion

The required quadratic polynomial is:

\[ \boxed{x^2 + 2\sqrt{3}x – 9} \]

The zeros of the polynomial are:

\[ \sqrt{3} \quad \text{and} \quad -3\sqrt{3} \]

\[ \therefore \quad \text{The required result is obtained.} \]