how to find the other two zeroes if two of them are already given for a 4 degree polynomial. example me with the help of an unique example. thanks

Suppose a and b are roots of a fourth degree polynomial.

Then, (x - a) and (x - b) will be the factors of the polynomial and hence, (x-a)(x-b) will be a factor of the polynomial.

Now, to find the other two factors, divide the polynomial with (x-a)(x-b) = x² - x(a+b) + ab. You will get 0 as the remainder and the quotient will be a second degree polynomial.

By division algorithm,

p(x) = q(x) (x-a)(x-b) 

Hence, the roots of q(x) will be the remaining two roots of the fourth degree polynomial.

 

Example: If two of the zeroes of the polynomial p(x) = 5x⁴ - 5x³ - 33x² + 3x + 18 are, find the other two zeroes.

 

Solution: