Polynomial Help (1 Viewer)

[Kujah]

I need some help please

Solve the equation x⁴ - 2x³ - 3x² + 4x + 2 = 0, given that the sum of two of its roots is zero.

 

[haque]

Let the roots be a,b,c,d then
a+b=0, a=-b and c+d=2(sum of roots)
c=2-d
and sum of roots three at a time is
abc+acd+bcd+abd=-4
-a^2c+acd+bcd-a^2d=-4
-a^2(c+d) +cd(a+b)=-4
-a^2(2)=-4 so a=sqrt2 and b=-sqrt2 then u know x^2-2 is a factor, divide the polynomial by that and u get something like the roots x=+-sqrt2, 1+-sqrt2

 

[Yip]

Alternative method:
By inspection, u can group it this way:
x⁴ - 2x³ - 3x² + 4x + 2 = 0
(x⁴ - 3x²+ 2)-2x(x² -2)=0
(x² -2)(x² -1)-2x(x² -2)=0
(x² -2)[x² -2x-1]=0
x=+-root2, 1+-root2