Find the Zeros of f(v) = v² + 4√3v − 15 and Verify the Relationship Between Zeros and Coefficients

Video Explanation

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Solution

Given polynomial:

f(v) = v² + 4√3v − 15

Step 1: Find the Zeros of the Polynomial

v² + 4√3v − 15 = 0

Split the middle term:

v² + 5√3v − √3v − 15 = 0

Grouping the terms:

v(v + 5√3) − √3(v + 5√3) = 0

(v + 5√3)(v − √3) = 0

∴ v + 5√3 = 0   or   v − √3 = 0

∴ v = −5√3   or   v = √3

Zeros of the polynomial are −5√3 and √3.

Step 2: Identify Coefficients

Comparing f(v) = v² + 4√3v − 15 with av² + bv + c:

a = 1,   b = 4√3,   c = −15

Step 3: Verify the Relationship

Let α = √3 and β = −5√3

Sum of zeros:

α + β = √3 + (−5√3) = −4√3

−b/a = −(4√3)/1 = −4√3

✔ Sum of zeros = −b/a

Product of zeros:

αβ = (√3)(−5√3) = −15

c/a = −15/1 = −15

✔ Product of zeros = c/a

Final Answer

Zeros of the polynomial are √3 and −5√3.

The relationship between zeros and coefficients is verified.

Conclusion

Thus, for the quadratic polynomial f(v) = v² + 4√3v − 15, the sum and product of zeros satisfy the standard relationships with its coefficients.