Question

find all zeros of zeros of polynomial 2 X raised to 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

Answer:

1 and -1/2

Step-by-step explanation:

Since 2 + √3 and 2 - √3 are roots of the polynomial, then by the Factor Theorem, the given polynomial is divisible by

( x - ( 2 + √3 ) ) ( x - ( 2 - √3 ) ) = x² - 4x + 1.

That is,

2x^4 - 9x³ + 5x² + 3x - 1 = ( x² - 4x + 1 ) ( a x² + b x + c )

for some a, b and c.

We could use the division algorithm here, or we can equate coefficients.  Let's do the latter.

The coefficient of x^4 is 2, and when we expand the right, the coefficient of x^4 is a.  So a = 2.

The constant coefficient is -1 on the left, and on the right it is c.  So c = -1.

The coefficient of x on the left is 3, and expanding the right, we find the coefficient of x is b - 4c.  So b - 4c = 3  =>  b = 3 + 4c = 3 - 4 = -1.

We could use the other two coefficients to check our answers, but let's just proceed confidently.

So the second quadratic factor is

a x² + b x + c = 2 x² - x - 1 = ( 2x + 1 ) ( x - 1 ).

The other roots of the original polynomial are then the roots of this quadratic, so they are 1 and -1/2.