Question

Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x),
where p(x) = x⁵ − 4x³ + x² + 3x +1, g(x) = x³ − 3x + 1

Answer 1

We have been given two polynomials

P(x) = x⁵ - 4x³ + x² + 3x + 1 and g(x) = x³ - 3x + 1

We will say g(x) is factor of p(x) if remainder is zero when we divide p(x) by g(x).

x3 -3x + 1)^{`("x"^2-1)/("x"^5-4"x"^3+"x"^2+3"x"+1)`}
                  `"x"^5-"x"^3+"x"^2`
                 -     +       -        
                -x³ + 3x +1 
                -x³ + 3x - 1                         
                +     -     +          
                                2       

Here, the remainder is 2 ≠ 0
g(x) is not a factor of p(x)