# 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.

**☛ Check: **NCERT Solutions Class 10 Maths Chapter 2

**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.

**☛ Related Questions:**

- Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients in each case:(i) 2x3 + x2 - 5x + 2; 1/2, 1, - 2(ii) x3 - 4x2 + 5x - 2; 2, 1, 1
- Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, - 7, - 14 respectively.
- If the zeroes of the polynomial x^3 - 3x^2 + x + 1 are a - b, a, a + b, find a and b.
- If the polynomial x4 - 6x3 + 16x2 - 25x + 10 is divided by another polynomial x2 - 2x + k, the remainder comes out to be x + a, find k and a.

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