Finding All Zeroes of a Polynomial

Video Explanation

Question

Find all the zeroes of the polynomial

\[ f(x) = 2x^4 – 2x^3 – 7x^2 + 3x + 6, \]

if two of its zeroes are \[ -\frac{\sqrt{3}}{2} \text{ and } \frac{\sqrt{3}}{2}. \]

Solution

Step 1: Form the Quadratic Factor from the Given Zeroes

Since the given zeroes are \[ -\frac{\sqrt{3}}{2} \text{ and } \frac{\sqrt{3}}{2}, \]

their corresponding quadratic factor is:

\[ \left(x – \frac{\sqrt{3}}{2}\right)\left(x + \frac{\sqrt{3}}{2}\right) = x^2 – \frac{3}{4} \]

Multiplying by 4 to remove fractions:

\[ 4x^2 – 3 \]

Hence, \(4x^2 – 3\) is a factor of the given polynomial.

Step 2: Divide the Polynomial by \(4x^2 – 3\)

Dividing

\[ 2x^4 – 2x^3 – 7x^2 + 3x + 6 \]

by

\[ 4x^2 – 3, \]

we get:

\[ 2x^4 – 2x^3 – 7x^2 + 3x + 6 = (4x^2 – 3)(x^2 – x – 2) \]

Step 3: Factorise the Remaining Quadratic Polynomial

\[ x^2 – x – 2 \]

\[ = (x – 2)(x + 1) \]

Step 4: Write the Complete Factorisation

\[ f(x) = (4x^2 – 3)(x – 2)(x + 1) \]

Step 5: Obtain All the Zeroes

Equating each factor to zero:

\[ 4x^2 – 3 = 0 \Rightarrow x = \pm \frac{\sqrt{3}}{2} \]

\[ x – 2 = 0 \Rightarrow x = 2 \]

\[ x + 1 = 0 \Rightarrow x = -1 \]

Conclusion

The zeroes of the polynomial

\[ 2x^4 – 2x^3 – 7x^2 + 3x + 6 \]

are

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