Find the Zeros of f(x) = x² − 2x − 8 and Verify the Relationship Between Zeros and Coefficients

Video Explanation

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Solution

Given polynomial:

f(x) = x² − 2x − 8

Step 1: Find the Zeros of the Polynomial

x² − 2x − 8 = 0

Split the middle term:

x² − 4x + 2x − 8 = 0

Grouping the terms:

x(x − 4) + 2(x − 4) = 0

(x − 4)(x + 2) = 0

∴ x − 4 = 0   or   x + 2 = 0

∴ x = 4   or   x = −2

Zeros of the polynomial are 4 and −2.

Step 2: Identify Coefficients

Comparing f(x) = x² − 2x − 8 with ax² + bx + c:

a = 1,   b = −2,   c = −8

Step 3: Verify the Relationship

Let α = 4 and β = −2

Sum of zeros:

α + β = 4 + (−2) = 2

−b/a = −(−2)/1 = 2

✔ Sum of zeros = −b/a

Product of zeros:

αβ = 4 × (−2) = −8

c/a = −8/1 = −8

✔ Product of zeros = c/a

Final Answer

Zeros of the polynomial are 4 and −2.

The relationship between zeros and coefficients is verified.

Conclusion

Thus, for the quadratic polynomial f(x) = x² − 2x − 8, the sum and product of zeros satisfy the standard relationships with its coefficients.