If α and β are the zeros of the quadratic polynomial f(x) = x² − 2x + 3, find a polynomial whose roots are α + 2 and β + 2

Video Explanation

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Solution

Given polynomial:

f(x) = x² − 2x + 3

Step 1: Find α + β and αβ

Comparing f(x) = x² − 2x + 3 with ax² + bx + c:

a = 1,   b = −2,   c = 3

α + β = −b/a = 2

αβ = c/a = 3

Step 2: Find the Sum and Product of New Roots

New roots are α + 2 and β + 2.

Sum of new roots:

(α + 2) + (β + 2) = (α + β) + 4 = 2 + 4 = 6

Product of new roots:

(α + 2)(β + 2)

= αβ + 2(α + β) + 4

= 3 + 2(2) + 4

= 11

Step 3: Form the Required Polynomial

The quadratic polynomial whose roots are α + 2 and β + 2 is:

x² − (sum of roots)x + (product of roots)

= x² − 6x + 11

Final Answer

The required polynomial is x² − 6x + 11.

Conclusion

Thus, the quadratic polynomial whose roots are α + 2 and β + 2 is x² − 6x + 11.