Finding All Zeroes of a Polynomial

Video Explanation

Question

Find all the zeroes of the polynomial

\[ f(x) = 2x^4 + 7x^3 – 19x^2 – 14x + 30, \]

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

\[ 2x^4 + 7x^3 – 19x^2 – 14x + 30 \]

by

\[ x^2 – 2, \]

we get:

\[ 2x^4 + 7x^3 – 19x^2 – 14x + 30 = (x^2 – 2)(2x^2 + 7x – 15) \]

Step 3: Factorise the Remaining Quadratic Polynomial

\[ 2x^2 + 7x – 15 \]

\[ = (2x – 3)(x + 5) \]

Step 4: Write the Complete Factorisation

\[ f(x) = (x^2 – 2)(2x – 3)(x + 5) \]

Step 5: Obtain All the Zeroes

Equating each factor to zero:

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

\[ 2x – 3 = 0 \Rightarrow x = \frac{3}{2} \]

\[ x + 5 = 0 \Rightarrow x = -5 \]

Conclusion

The zeroes of the polynomial

\[ 2x^4 + 7x^3 – 19x^2 – 14x + 30 \]

are

\[ \boxed{-\sqrt{2},\; \sqrt{2},\; \frac{3}{2},\; -5} \]