Find the Zeros of g(x) = a(x² + 1) − x(a² + 1) and Verify the Relationship Between Zeros and Coefficients

Video Explanation

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Solution

Given polynomial:

g(x) = a(x² + 1) − x(a² + 1)

Expanding the terms:

g(x) = ax² − (a² + 1)x + a

Step 1: Find the Zeros of the Polynomial

ax² − (a² + 1)x + a = 0

Split the middle term:

ax² − a²x − x + a = 0

Grouping the terms:

ax(x − a) − 1(x − a) = 0

(x − a)(ax − 1) = 0

∴ x − a = 0   or   ax − 1 = 0

∴ x = a   or   x = 1/a

Zeros of the polynomial are a and 1/a.

Step 2: Identify Coefficients

Comparing g(x) = ax² − (a² + 1)x + a with ax² + bx + c:

a = a,   b = −(a² + 1),   c = a

Step 3: Verify the Relationship

Let α = a and β = 1/a

Sum of zeros:

α + β = a + 1/a

−b/a = −[−(a² + 1)] / a = (a² + 1)/a = a + 1/a

✔ Sum of zeros = −b/a

Product of zeros:

αβ = a × (1/a) = 1

c/a = a/a = 1

✔ Product of zeros = c/a

Final Answer

Zeros of the polynomial are a and 1/a.

The relationship between zeros and coefficients is verified.

Conclusion

Thus, for the quadratic polynomial g(x) = a(x² + 1) − x(a² + 1), the sum and product of zeros satisfy the standard relationships with its coefficients.