Find all zeros of the polynomial 2x⁴ + 7x³ − 19x² − 14x + 30, if two of its zeroes are √2 and −√2

Video Explanation

Watch the video explanation below:

Given

f(x) = 2x⁴ + 7x³ − 19x² − 14x + 30

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 Corresponding to the Given Zeroes

Since √2 and −√2 are zeroes, their corresponding factor is:

(x − √2)(x + √2)

= x² − (√2)²

= x² − 2

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

Divide 2x⁴ + 7x³ − 19x² − 14x + 30 by x² − 2:

First term:

2x⁴ ÷ x² = 2x²

Multiply:

2x²(x² − 2) = 2x⁴ − 4x²

Subtract:

(2x⁴ + 7x³ − 19x²) − (2x⁴ − 4x²)

= 7x³ − 15x²

Bring down −14x:

7x³ − 15x² − 14x

Next term:

7x³ ÷ x² = 7x

Multiply:

7x(x² − 2) = 7x³ − 14x

Subtract:

(7x³ − 15x² − 14x) − (7x³ − 14x)

= −15x²

Bring down +30:

−15x² + 30

Next term:

−15x² ÷ x² = −15

Multiply:

−15(x² − 2) = −15x² + 30

Subtract:

(−15x² + 30) − (−15x² + 30) = 0

Remainder is zero, hence x² − 2 is a factor.

Quotient obtained:

2x² + 7x − 15

Step 3: Factorise the Quadratic Polynomial

2x² + 7x − 15 = 0

Splitting the middle term:

2x² + 10x − 3x − 15 = 0

2x(x + 5) − 3(x + 5) = 0

(2x − 3)(x + 5) = 0

∴ x = 3/2 or x = −5

Final Answer

All the zeroes of the given polynomial are:

√2, −√2, 3/2 and −5

Conclusion

Thus, the polynomial 2x⁴ + 7x³ − 19x² − 14x + 30 has four zeroes: √2, −√2, 3/2 and −5.