Find all zeros of the polynomial 2x⁴ − 9x³ + 5x² + 3x − 1, if two of its zeroes are 2 + √3 and 2 − √3

Video Explanation

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Given

f(x) = 2x⁴ − 9x³ + 5x² + 3x − 1

Two zeroes of the polynomial are:

x = 2 + √3 and x = 2 − √3

To Find

All the zeroes of the given polynomial.

Solution

Step 1: Form the Factor Using Given Zeroes

Since 2 + √3 and 2 − √3 are zeroes, the corresponding factor is:

(x − (2 + √3))(x − (2 − √3))

= [(x − 2) − √3][(x − 2) + √3]

= (x − 2)² − (√3)²

= (x − 2)² − 3

= x² − 4x + 1

Step 2: Divide the Polynomial by (x² − 4x + 1)

Divide 2x⁴ − 9x³ + 5x² + 3x − 1 by x² − 4x + 1:

First term:

2x⁴ ÷ x² = 2x²

Multiply:

2x²(x² − 4x + 1) = 2x⁴ − 8x³ + 2x²

Subtract:

(2x⁴ − 9x³ + 5x²) − (2x⁴ − 8x³ + 2x²)

= −x³ + 3x²

Bring down +3x:

−x³ + 3x² + 3x

Next term:

−x³ ÷ x² = −x

Multiply:

−x(x² − 4x + 1) = −x³ + 4x² − x

Subtract:

(−x³ + 3x² + 3x) − (−x³ + 4x² − x)

= −x² + 4x

Bring down −1:

−x² + 4x − 1

Next term:

−x² ÷ x² = −1

Multiply:

−1(x² − 4x + 1) = −x² + 4x − 1

Subtract:

(−x² + 4x − 1) − (−x² + 4x − 1) = 0

Remainder is zero.

Quotient obtained:

2x² − x − 1

Step 3: Factorise the Quadratic Polynomial

2x² − x − 1 = 0

Splitting the middle term:

2x² − 2x + x − 1 = 0

2x(x − 1) + 1(x − 1) = 0

(2x + 1)(x − 1) = 0

∴ x = −1/2 or x = 1

Final Answer

All the zeroes of the given polynomial are:

2 + √3, 2 − √3, −1/2 and 1

Conclusion

Thus, the polynomial 2x⁴ − 9x³ + 5x² + 3x − 1 has four zeroes: 2 + √3, 2 − √3, −1/2 and 1.