Obtain all zeros of the polynomial f(x) = x⁴ − 3x³ − x² + 9x − 6, if two of its zeros are −√3 and √3

Video Explanation

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Given

f(x) = x⁴ − 3x³ − x² + 9x − 6

Two zeroes of the polynomial are:

x = −√3 and x = √3

To Find

All the zeroes of the given polynomial.

Solution

Step 1: Use the Fact That ±√3 Are Zeroes

Since −√3 and √3 are zeroes of f(x), their corresponding factor is:

(x − √3)(x + √3) = x² − (√3)² = x² − 3

Step 2: Divide f(x) by x² − 3

We divide the polynomial x⁴ − 3x³ − x² + 9x − 6 by x² − 3.

First term:

x⁴ ÷ x² = x²

Multiply:

x²(x² − 3) = x⁴ − 3x²

Subtract:

(x⁴ − 3x³ − x²) − (x⁴ − 3x²) = −3x³ + 2x²

Bring down next terms:

−3x³ + 2x² + 9x − 6

Next term:

−3x³ ÷ x² = −3x

Multiply:

−3x(x² − 3) = −3x³ + 9x

Subtract:

(−3x³ + 2x² + 9x) − (−3x³ + 9x) = 2x²

Bring down −6:

2x² − 6

Next term:

2x² ÷ x² = 2

Multiply:

2(x² − 3) = 2x² − 6

Subtract:

(2x² − 6) − (2x² − 6) = 0

So remainder = 0.

Quotient = x² − 3x + 2

Step 3: Find the Remaining Zeroes

Now solve the quadratic x² − 3x + 2 = 0

Factorising:

x² − 3x + 2 = (x − 1)(x − 2)

∴ x = 1 or x = 2

Final Answer

All the zeroes of the given polynomial are:

−√3, √3, 1 and 2

Conclusion

Thus, the polynomial f(x) = x⁴ − 3x³ − x² + 9x − 6 has four zeroes: −√3, √3, 1 and 2.