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Given that x - √5 is a factor of the cubic polynomial x³ - 3√5x² + 13x - 3√5, find all the zeroes of the polynomial

Solution:

Give, the cubic polynomial p(x) = x³ - 3√5x² + 13x - 3√5

g(x) = x - √5

We have to find all the zeros of the polynomial.

By using long division,

Given that x - √5 is a factor of the cubic polynomial x³ - 3√5x² + 13x - 3√5, find all the zeroes of the polynomial

The quotient is g(x) = x² - 2√5x + 3

On factoring,

x² - 2√5x + 3 = 0

Using the quadratic formula,

x= [-b ± √b² - 4ac]/2a

Here, a = 1, b = -2√5 and c = 3

x = [2√5 ± √(-2√5)² - 4 (1) (3)]/ 2(1)

x = [2√5 ± √20 - 12]/ 2

x = [2√5 ± √8]/ 2

x = [2√5 ± 2√2]/ 2

Taking out common terms,

x = 2[√5 ± √2]/ 2

x = √5 ± √2

Therefore, the zeros are √5, √5 ± √2.

✦ Try This: Given that x - (√5+√2) is a factor of the cubic polynomial x³ - 3√5x² + 13x - 3√5 , find all the zeroes of the polynomial

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 2

NCERT Exemplar Class 10 Maths Exercise 2.4 Problem 5

Summary:

Given that x - √5 is a factor of the cubic polynomial x³ - 3√5x² + 13x - 3√5, all the zeroes of the polynomial are √5, √5 ± √2.

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