Finding All Zeroes of a Cubic Polynomial

Video Explanation

Question

Find all the zeroes of the polynomial

\[ f(x) = 2x^3 + x^2 – 6x – 3, \]

if two of its zeroes are \( -\sqrt{3} \) and \( \sqrt{3} \).

Solution

Step 1: Form the Quadratic Factor from the Given Zeroes

Since the given zeroes are \( -\sqrt{3} \) and \( \sqrt{3} \),

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

Hence, \(x^2 – 3\) is a factor of the given polynomial.

Step 2: Divide the Polynomial by \(x^2 – 3\)

Dividing

\[ 2x^3 + x^2 – 6x – 3 \]

by

\[ x^2 – 3, \]

we get:

\[ 2x^3 + x^2 – 6x – 3 = (x^2 – 3)(2x + 1) \]

Step 3: Write the Complete Factorisation

\[ f(x) = (x^2 – 3)(2x + 1) \]

Step 4: Obtain All the Zeroes

Equating each factor to zero:

\[ x^2 – 3 = 0 \Rightarrow x = \pm \sqrt{3} \]

\[ 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \]

Conclusion

The zeroes of the polynomial

\[ 2x^3 + x^2 – 6x – 3 \]

are

\[ \boxed{-\sqrt{3},\; \sqrt{3},\; -\frac{1}{2}} \]