Find a Quadratic Polynomial Whose Sum and Product of Zeros Are −8/3 and 4/3 and Find Its Zeros by Factorisation

Video Explanation

Watch the video explanation below:

Solution

Given:

Sum of zeros = −8/3
Product of zeros = 4/3

Step 1: Form the Quadratic Polynomial

If sum of zeros = α + β and product of zeros = αβ, then the quadratic polynomial is:

x² − (α + β)x + αβ

∴ Required polynomial:

x² − (−8/3)x + 4/3

x² + (8/3)x + 4/3

Multiplying throughout by 3 to remove fractions:

3x² + 8x + 4

Step 2: Find the Zeros by Factorisation

3x² + 8x + 4 = 0

Split the middle term:

3x² + 6x + 2x + 4 = 0

Grouping the terms:

3x(x + 2) + 2(x + 2) = 0

(3x + 2)(x + 2) = 0

∴ 3x + 2 = 0   or   x + 2 = 0

∴ x = −2/3   or   x = −2

Zeros of the polynomial are −2/3 and −2.

Step 3: Verification

Let α = −2/3 and β = −2

Sum of zeros:

α + β = −2/3 + (−2) = −8/3

✔ Verified

Product of zeros:

αβ = (−2/3)(−2) = 4/3

✔ Verified

Final Answer

Required quadratic polynomial is 3x² + 8x + 4.

Zeros of the polynomial are −2/3 and −2.

Conclusion

Thus, the quadratic polynomial formed using the given sum and product of zeros satisfies all the required conditions, and its zeros are obtained correctly by the factorisation method.