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

Solution:

Given polynomial is x⁴ - 6x³ - 26x² + 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) = x⁴ - 6x³ - 26x² + 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

= x² - 4x + 1

Thus, x² - 4x + 1 is a factor of the given polynomial

Now, let's divide x⁴ - 6x³ - 26x² + 138x - 35 by x² - 4x + 1

Clearly, by division algorithm,

x⁴ - 6x³ - 26x² + 138x - 35 = (x² - 4x + 1)(x² - 2x - 35)

It can be observed that x² - 2x - 35 is a factor of the given polynomial

Also, x² - 2x - 35 = x² - 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 x⁴ - 6x³ - 26x² + 138x - 35 are 2 ± √3, the other zeros are 7 and -5.

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