Finding All Zeroes of a Cubic Polynomial

Video Explanation

Question

Find all the zeroes of the polynomial

\[ f(x) = 3x^3 + 10x^2 – 9x – 4, \]

if one of its zeroes is \(1\).

Solution

Step 1: Verify the Given Zero

Substitute \(x = 1\) in \(f(x)\):

\[ f(1) = 3(1)^3 + 10(1)^2 – 9(1) – 4 \]

\[ = 3 + 10 – 9 – 4 = 0 \]

Hence, \(1\) is a zero of the polynomial.

Step 2: Divide the Polynomial by \((x – 1)\)

Since \(1\) is a zero, \((x – 1)\) is a factor of the polynomial.

Dividing \(3x^3 + 10x^2 – 9x – 4\) by \((x – 1)\), we get:

\[ 3x^3 + 10x^2 – 9x – 4 = (x – 1)(3x^2 + 13x + 4) \]

Step 3: Find the Remaining Zeroes

Now, factorise the quadratic polynomial:

\[ 3x^2 + 13x + 4 \]

\[ = (3x + 1)(x + 4) \]

So,

\[ 3x^3 + 10x^2 – 9x – 4 = (x – 1)(3x + 1)(x + 4) \]

Step 4: Write All the Zeroes

Equating each factor to zero:

\[ x – 1 = 0 \Rightarrow x = 1 \]

\[ 3x + 1 = 0 \Rightarrow x = -\frac{1}{3} \]

\[ x + 4 = 0 \Rightarrow x = -4 \]

Conclusion

The zeroes of the polynomial \[ 3x^3 + 10x^2 – 9x – 4 \] are

\[ \boxed{1,\; -\frac{1}{3},\; -4} \]