If the product of two zeros of the polynomial f(x) = 2x³ + 6x² − 4x + 9 is 3, find its third zero

Video Explanation

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Given

f(x) = 2x³ + 6x² − 4x + 9

Product of two zeros = 3

To Find

The third zero of the polynomial.

Solution

For a cubic polynomial:

ax³ + bx² + cx + d

Product of all three zeros = −d / a

Step 1: Find the Product of All Zeros

Comparing f(x) = 2x³ + 6x² − 4x + 9 with ax³ + bx² + cx + d,

a = 2,   d = 9

Product of all zeros = −d / a

= −9 / 2

Step 2: Use the Given Information

Let the three zeros be α, β and γ.

Given:

αβ = 3

αβγ = −9 / 2

Step 3: Find the Third Zero

γ = (αβγ) / (αβ)

= (−9 / 2) ÷ 3

= −9 / 6

= −3 / 2

Final Answer

The third zero of the polynomial is −3/2.

Conclusion

Hence, if the product of two zeros of the polynomial f(x) = 2x³ + 6x² − 4x + 9 is 3, then the third zero is −3/2.