Finding the Third Zero of a Cubic Polynomial

Video Explanation

Question

If two zeroes of the polynomial

\[ f(x) = x^3 + 3x^2 – 5x – 15 \]

are \( \sqrt{5} \) and \( -\sqrt{5} \), find its third zero.

Options:

(a) 3    (b) -3    (c) 5    (d) -5

Solution

Step 1: Use the Formula for Sum of Zeroes

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

the sum of its zeroes is:

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

Step 2: Apply to the Given Polynomial

Given:

\[ f(x) = x^3 + 3x^2 – 5x – 15 \]

Here,

\[ a = 1, \quad b = 3 \]

So, the sum of all three zeroes is:

\[ -\frac{3}{1} = -3 \]

Step 3: Use the Given Zeroes

Sum of the two given zeroes:

\[ \sqrt{5} + (-\sqrt{5}) = 0 \]

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

Then,

\[ 0 + \alpha = -3 \]

\[ \alpha = -3 \]

Conclusion

The third zero of the polynomial is:

\[ \boxed{-3} \]

Hence, the correct option is (b) -3.