# If two zeroes of the polynomial x^{4} - 6x^{3} - 26x^{2} + 138x - 35 are 2 ± √3, find other zeroes.

**Solution:**

Given polynomial is x^{4} - 6x^{3} - 26x^{2} + 138x - 35 and the zeroes of the polynomial are 2 ± √3

- By using the zeroes of a polynomial, we can find out the factors of the polynomial.
- Now let's divide the polynomial with the factor to get the quotient and remainder.
- Substitute this value in the division algorithm to get the other zeroes by simplifying its factors.

p (x) = x^{4} - 6x^{3} - 26x^{2} + 138x - 35

Zeroes of the polynomial are 2 ± √3

Thus the factors are (x - 2 + √3) and (x - 2 - √3)

Therefore,

(x - 2 + √3)(x - 2 - √3) = x² + 4 - 4x - 3

= x^{2} - 4x + 1

Thus, x^{2} - 4x + 1 is a factor of the given polynomial

Now, let's divide x^{4} - 6x^{3} - 26x^{2} + 138x - 35 by x^{2} - 4x + 1

Clearly, by division algorithm,

x^{4} - 6x^{3} - 26x^{2} + 138x - 35 = (x^{2} - 4x + 1)(x^{2} - 2x - 35)

It can be observed that x^{2} - 2x - 35 is a factor of the given polynomial

Also, x^{2} - 2x - 35 = x^{2} - 7x + 5x - 35 = (x - 7) (x + 5)

Therefore, the value of the polynomial is also zero when x - 7 = 0 or x + 5 = 0

Hence, 7 and - 5 are also zeroes of this polynomial.

**Video Solution:**

## If two zeroes of the polynomial x⁴ - 6x³ - 26x² + 138x - 35 are 2 ± √3 find other zeroes.

### NCERT Solutions Class 10 Maths - Chapter 2 Exercise 2.4 Question 4

**Summary:**

If two zeroes of the polynomial x^{4} - 6x^{3} - 26x^{2} + 138x - 35 are 2 ± √3, the other zeros are 7 and -5.