Finding All Zeroes of a Polynomial

Video Explanation

Question

Find all the zeroes of the polynomial

\[ f(x) = x^4 + x^3 – 34x^2 – 4x + 120, \]

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

Solution

Step 1: Form the Quadratic Factor from Given Zeroes

Since \(2\) and \(-2\) are zeroes,

\[ (x – 2)(x + 2) = x^2 – 4 \]

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

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

Dividing

\[ x^4 + x^3 – 34x^2 – 4x + 120 \]

by

\[ x^2 – 4, \]

we get:

\[ x^4 + x^3 – 34x^2 – 4x + 120 = (x^2 – 4)(x^2 + x – 30) \]

Step 3: Factorise the Remaining Quadratic Polynomial

\[ x^2 + x – 30 \]

\[ = (x + 6)(x – 5) \]

Step 4: Write the Complete Factorisation

\[ f(x) = (x – 2)(x + 2)(x + 6)(x – 5) \]

Step 5: Obtain All the Zeroes

Equating each factor to zero:

\[ x – 2 = 0 \Rightarrow x = 2 \]

\[ x + 2 = 0 \Rightarrow x = -2 \]

\[ x + 6 = 0 \Rightarrow x = -6 \]

\[ x – 5 = 0 \Rightarrow x = 5 \]

Conclusion

The zeroes of the polynomial

\[ x^4 + x^3 – 34x^2 – 4x + 120 \]

are

\[ \boxed{2,\; -2,\; -6,\; 5} \]