Finding the Third Zero of a Cubic Polynomial

Video Explanation

Question

If the product of two zeroes of the polynomial

\[ f(x) = 2x^3 + 6x^2 – 4x + 9 \]

is \(3\), find its third zero.

Solution

Step 1: Use the Formula for Product of Zeroes

For a cubic polynomial \[ ax^3 + bx^2 + cx + d, \]

the product of all three zeroes is given by

\[ -\frac{d}{a} \]

Step 2: Apply to the Given Polynomial

Here,

\[ a = 2, \quad d = 9 \]

So, product of all three zeroes:

\[ \alpha \beta \gamma = -\frac{9}{2} \]

Step 3: Find the Third Zero

Given that the product of two zeroes is \(3\).

Let the third zero be \(\gamma\).

\[ 3 \times \gamma = -\frac{9}{2} \]

\[ \gamma = -\frac{9}{6} = -\frac{3}{2} \]

Conclusion

The third zero of the given polynomial is:

\[ \boxed{-\frac{3}{2}} \]