Finding All Zeroes of a Cubic Polynomial

Video Explanation

Question

Obtain all the zeroes of the polynomial

\[ f(x) = x^3 + 13x^2 + 32x + 20, \]

if one of its zeroes is \(-2\).

Solution

Step 1: Use the Given Zero

Since \(-2\) is a zero of the polynomial, \((x + 2)\) is a factor of \(f(x)\).

Step 2: Divide the Polynomial by \((x + 2)\)

Dividing \[ x^3 + 13x^2 + 32x + 20 \] by \[ x + 2, \] we get:

\[ x^3 + 13x^2 + 32x + 20 = (x + 2)(x^2 + 11x + 10) \]

Step 3: Factorise the Quadratic Polynomial

\[ x^2 + 11x + 10 \]

\[ = (x + 1)(x + 10) \]

Step 4: Write All the Factors

\[ f(x) = (x + 2)(x + 1)(x + 10) \]

Step 5: Obtain All the Zeroes

Equating each factor to zero:

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

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

\[ x + 10 = 0 \Rightarrow x = -10 \]

Conclusion

The zeroes of the polynomial

\[ x^3 + 13x^2 + 32x + 20 \]

are

\[ \boxed{-2,\; -1,\; -10} \]