Find all the zeros of the polynomial 2x³ + x² − 6x − 3, if two of its zeroes are −√3 and √3

Video Explanation

Watch the video explanation below:

Given

f(x) = 2x³ + x² − 6x − 3

Two zeroes of the polynomial are:

x = −√3 and x = √3

To Find

All the zeroes of the given polynomial.

Solution

Step 1: Form the Factor Using Given Zeroes

Since −√3 and √3 are zeroes of the polynomial, the corresponding factor is:

(x − √3)(x + √3)

= x² − (√3)²

= x² − 3

Step 2: Divide the Polynomial by (x² − 3)

Divide 2x³ + x² − 6x − 3 by x² − 3:

First term:

2x³ ÷ x² = 2x

Multiply:

2x(x² − 3) = 2x³ − 6x

Subtract:

(2x³ + x² − 6x) − (2x³ − 6x)

= x²

Bring down −3:

x² − 3

Next term:

x² ÷ x² = 1

Multiply:

1(x² − 3) = x² − 3

Subtract:

(x² − 3) − (x² − 3) = 0

So remainder is zero.

Quotient obtained:

2x + 1

Step 3: Find the Remaining Zero

Now solve:

2x + 1 = 0

∴ x = −1/2

Final Answer

All the zeroes of the given polynomial are:

−√3, √3 and −1/2

Conclusion

Hence, the polynomial 2x³ + x² − 6x − 3 has three zeroes: −√3, √3 and −1/2.