Finding All Zeroes of a Cubic Polynomial

Video Explanation

Question

Find all the zeroes of the polynomial

\[ f(x) = x^3 + 3x^2 – 2x – 6, \]

if two of its zeroes are \( -\sqrt{2} \) and \( \sqrt{2} \).

Solution

Step 1: Form the Quadratic Factor from the Given Zeroes

Since the given zeroes are \( -\sqrt{2} \) and \( \sqrt{2} \),

\[ (x – \sqrt{2})(x + \sqrt{2}) = x^2 – 2 \]

Hence, \(x^2 – 2\) is a factor of the given polynomial.

Step 2: Divide the Polynomial by \(x^2 – 2\)

Dividing

\[ x^3 + 3x^2 – 2x – 6 \]

by

\[ x^2 – 2, \]

we get:

\[ x^3 + 3x^2 – 2x – 6 = (x^2 – 2)(x + 3) \]

Step 3: Write the Complete Factorisation

\[ f(x) = (x^2 – 2)(x + 3) \]

Step 4: Obtain All the Zeroes

Equating each factor to zero:

\[ x^2 – 2 = 0 \Rightarrow x = \pm \sqrt{2} \]

\[ x + 3 = 0 \Rightarrow x = -3 \]

Conclusion

The zeroes of the polynomial

\[ x^3 + 3x^2 – 2x – 6 \]

are

\[ \boxed{-\sqrt{2},\; \sqrt{2},\; -3} \]