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

Video Explanation

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Given

f(x) = x³ + 3x² − 2x − 6

Two zeroes of the polynomial are:

x = −√2 and x = √2

To Find

All the zeroes of the given polynomial.

Solution

Step 1: Form the Factor from the Given Zeroes

Since −√2 and √2 are zeroes of the polynomial, the corresponding factor is:

(x − √2)(x + √2)

= x² − (√2)²

= x² − 2

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

Divide x³ + 3x² − 2x − 6 by x² − 2:

First term:

x³ ÷ x² = x

Multiply:

x(x² − 2) = x³ − 2x

Subtract:

(x³ + 3x² − 2x) − (x³ − 2x)

= 3x²

Bring down −6:

3x² − 6

Next term:

3x² ÷ x² = 3

Multiply:

3(x² − 2) = 3x² − 6

Subtract:

(3x² − 6) − (3x² − 6) = 0

So the remainder is zero.

Quotient obtained:

x + 3

Step 3: Find the Remaining Zero

Solve:

x + 3 = 0

∴ x = −3

Final Answer

All the zeroes of the given polynomial are:

−√2, √2 and −3

Conclusion

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