Finding Other Zeroes of a Cubic Polynomial

Video Explanation

Question

Given that \( \sqrt{2} \) is a zero of the cubic polynomial

\[ f(x) = 6x^3 + \sqrt{2}x^2 – 10x – 4\sqrt{2}, \]

find its other two zeroes.

Solution

Step 1: Use the Conjugate Root Theorem

Since the coefficients of the polynomial are real and \( \sqrt{2} \) is a zero, its conjugate \( -\sqrt{2} \) is also a zero.

Hence,

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

is a factor of the given polynomial.

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

Dividing

\[ 6x^3 + \sqrt{2}x^2 – 10x – 4\sqrt{2} \]

by

\[ x^2 – 2, \]

we get:

\[ 6x^3 + \sqrt{2}x^2 – 10x – 4\sqrt{2} = (x^2 – 2)(6x + \sqrt{2}) \]

Step 3: Obtain All the Zeroes

Equating each factor to zero:

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

\[ 6x + \sqrt{2} = 0 \Rightarrow x = -\frac{\sqrt{2}}{6} \]

Conclusion

The zeroes of the given polynomial are:

\[ \boxed{\sqrt{2},\; -\sqrt{2},\; -\frac{\sqrt{2}}{6}} \]