Find the Zeros of f(x) = x² − (√3 + 1)x + √3 and Verify the Relationship Between Zeros and Coefficients

Video Explanation

Watch the video explanation below:

Solution

Given polynomial:

f(x) = x² − (√3 + 1)x + √3

Step 1: Find the Zeros of the Polynomial

x² − (√3 + 1)x + √3 = 0

Split the middle term:

x² − √3x − x + √3 = 0

Grouping the terms:

x(x − √3) − 1(x − √3) = 0

(x − √3)(x − 1) = 0

∴ x − √3 = 0   or   x − 1 = 0

∴ x = √3   or   x = 1

Zeros of the polynomial are √3 and 1.

Step 2: Identify Coefficients

Comparing f(x) = x² − (√3 + 1)x + √3 with ax² + bx + c:

a = 1,   b = −(√3 + 1),   c = √3

Step 3: Verify the Relationship

Let α = √3 and β = 1

Sum of zeros:

α + β = √3 + 1

−b/a = −[−(√3 + 1)] / 1 = √3 + 1

✔ Sum of zeros = −b/a

Product of zeros:

αβ = (√3)(1) = √3

c/a = √3 / 1 = √3

✔ Product of zeros = c/a

Final Answer

Zeros of the polynomial are √3 and 1.

The relationship between zeros and coefficients is verified.

Conclusion

Thus, for the quadratic polynomial f(x) = x² − (√3 + 1)x + √3, the sum and product of zeros satisfy the standard relationships with its coefficients.