word problem

Question

proof that for any real number k, (x-k) is not a factor of x^4+x^2+1

Answer 1

x⁴+x²+1

let n=x²

n²+n+1

using the quadratic equation to factor this polynomial

 

[-1+√(1-4)]/2

[-1+√-3]/2

[-1+i√3]/2

(n+[1-i√3]/2)(n+[1+i√3]/2)

(x²+[1-i√3]/2)(x²+[1+i√3]/2)

looking at this, the polynomial x⁴+x²+1 can't be factored into two binomials with Real coefficients

because you see the imaginary numbers (i√3)

k any REAL number, (x-k) is a binomial with Real coefficients

if x⁴+x²+1=0, this polynomial would have only imaginary roots and k is a Real root