Question

obtain all other zeros of p(x)=x^4-6x^3-26x^2+138x-35, two zeroes are 2-√3and 2+√3​

Answer 1

alpha=2-√3

beta=2+√3

so, using formula

(x-alpha)(x-beta)

(x-(2-√3))(x-(2+√3)

(x-2+√3)(x-2-√3)

(x-2)^2-(√3)^2

x^2+2^2-2*x*2-3

x^2+4-4x-3

x^2-4x+1

dividing x^4-6x^3-26x^2+138x-35 by x^2-4x+1

we get quotient=x^2-2x-35

by splitting method

x^2-2x-35

x^2-7x+5x-35

x(x-7)+5(x-7)

(x-7)(x+5)

x=7,-5

hence two zeros are 7and-5