Question

find all the zeros of the polynomial 2x^4-9x^3+5x^2+3x-1 if two of its zeroes are (2+√3)and(2-√3).

Answer 1

Hola there..

According to the question,

Two zeroes of polynomial 2x⁴ - 9x³ + 5x² + 3x - 1 are (2 + √3) and (2 - √3)

We can write the zeroes as:

[x - (2 + √3)][x - (2 - √3)]

=> [x - 2 - √3][x - 2 + √3]

=> [x - 2]² - [√3]²

=> x² + 4 - 4x - 3

=> x² - 4x + 1

When we will divide this polynomial with p(x) then we will get other two zeroes.

=> x² - 4x + 1 )2x⁴ - 9x³ + 5x² +3x - 1 (2x² - x - 1
±2x⁴ ± 8x³ ± 2x²
----------------------------------
- x³ + 3x² + 3x - 1
±x³ ± 4x² ± x
-----------------------------
-x² + 4x - 1
±x² ± 4x ± 1
------------------------
0

So, we will get other two zeroes as;

=> 2x² - x - 1

=> 2x² - 2x + x - 1

=> 2x(x - 1) + 1(x - 1)

=> (2x + 1)(x - 1)

=> x = -1/2 and 1

Therefore, zeroes of polynomial

2x⁴ - 9x³ + 5x² + 3x - 1 = (2 + √3), (2 - √3), 1 and -1/2

Hope this helps....:)

Answer 2

Answer:

Hope this helps you ✌️✌️