Find the Zeros of g(s) = 4s² − 4s + 1 and Verify the Relationship Between Zeros and Coefficients

Video Explanation

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Solution

Given polynomial:

g(s) = 4s² − 4s + 1

Step 1: Find the Zeros of the Polynomial

4s² − 4s + 1 = 0

Split the middle term:

4s² − 2s − 2s + 1 = 0

Grouping the terms:

2s(2s − 1) − 1(2s − 1) = 0

(2s − 1)(2s − 1) = 0

(2s − 1)² = 0

∴ 2s − 1 = 0

∴ s = 1/2

Zeros of the polynomial are 1/2 and 1/2.

Step 2: Identify Coefficients

Comparing g(s) = 4s² − 4s + 1 with as² + bs + c:

a = 4,   b = −4,   c = 1

Step 3: Verify the Relationship

Let α = 1/2 and β = 1/2

Sum of zeros:

α + β = 1/2 + 1/2 = 1

−b/a = −(−4)/4 = 1

✔ Sum of zeros = −b/a

Product of zeros:

αβ = (1/2)(1/2) = 1/4

c/a = 1/4

✔ Product of zeros = c/a

Final Answer

Zeros of the polynomial are 1/2 and 1/2.

The relationship between zeros and coefficients is verified.

Conclusion

Thus, for the quadratic polynomial g(s) = 4s² − 4s + 1, the sum and product of zeros satisfy the standard relationships with its coefficients.