Polynomial from Given Zeros

Video Explanation

Question

If \( \alpha \) and \( \beta \) are the zeros of the quadratic polynomial

\[ f(x) = x^2 – 2x + 3, \]

find a polynomial whose zeros are

\[ \alpha + 2 \quad \text{and} \quad \beta + 2. \]

Solution

Step 1: Find Sum and Product of \( \alpha \) and \( \beta \)

Comparing \(x^2 – 2x + 3\) with \(ax^2 + bx + c\),

\[ a = 1,\quad b = -2,\quad c = 3 \]

\[ \alpha + \beta = -\frac{b}{a} = 2 \]

\[ \alpha\beta = \frac{c}{a} = 3 \]

Step 2: Find Sum and Product of New Zeros

New zeros are \( \alpha + 2 \) and \( \beta + 2 \).

Sum:

\[ (\alpha + 2) + (\beta + 2) = (\alpha + \beta) + 4 = 2 + 4 = 6 \]

Product:

\[ (\alpha + 2)(\beta + 2) = \alpha\beta + 2(\alpha + \beta) + 4 \]

\[ = 3 + 2(2) + 4 = 11 \]

Step 3: Form the Required Polynomial

A polynomial whose zeros have sum \(S\) and product \(P\) is:

\[ x^2 – Sx + P \]

\[ = x^2 – 6x + 11 \]

Conclusion

The required polynomial is:

\[ \boxed{x^2 – 6x + 11} \]

\[ \therefore \quad x^2 – 6x + 11 \text{ is the required polynomial.} \]