Question

find all the zeros of the polynomial 2 x to the power 4 - 9 x cube + 5 x square + 3 x minus 1 if two of its zeros are 2 + root 3 and 2 minus root 3 ​

Answer 1

Question : Find all the zeroes of Polynomial 2x⁴ - 9x³ + 5x² + 3x - 1. If two of its zeroes are 2 + √3 and 2 - √3.

SOLUTION :

Zeroes of polynomial = 2 + √3 and 2 - √3

x = 2 + √3 and x = 2 - √3

x - 2 + √3 = 0 and x - 2 - √3 = 0

[(x - 2) + √3] [(x - 2) - √3] = 0

= a² - b²

= (x - 2)² - 3

= x² - 2(x)(2) + 2² - 3

= x² - 4x + 4 - 3

= x² - 4x + 1

Now, Divide the polynomial with x² - 4x + 1

2x² - x - 1 is another factor of the polynomial.

Now,

Split the middle the terms.

=> 2x² - x - 1

=> 2x² - 2x + x - 1

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

=> x - 1 = 0 ; 2x + 1 = 0

=> x = 1 ; 2x = -1 = -1/2

Therefore, the other zeroes are -1/2 and 1.

Answer 2

Answer :-

1 and -1/2.

Given :-

$p(x) = 2x^4 -9x^3 + 5x^2 +3x -1$

Roots are :-

$2 + \sqrt{3}$

$2 -\sqrt{3}$

To find :-

It's all zeroes.

Solution:-

Steps to find the other two zeroes of the polynomial :-

• First find the g(x) by its given roots .

• Now , divide it by the given polynomial.

• The remainder becomes 0.

• Now the quoteint left is your required polynomial by which zeors can be obtained.

• Use middle term splliting method to factories it .

• Now equate with zero.

• You will get required to zeroes .

hence, the required to zeroes of the given polynomial are -1 and -1/2.