Question

Is x + 1 a factor or x^100 - 2x^50 - 4x^25 + 3x? Justify your answer. Please tell me how to solve it. Thanks!

Answer 1

in Factor  x^100-2x^50-4x^25+3x,If x+1 is a factor then the equation is zero , where x =-1,Now, Substituting the value of x in the equation  x^100-2x^50-4x^25+3x if the ans is zero then and only x+1 is said to be a factor (Proved)

Again On Factorising x^100-2x^50-4x^25+3x we get

x^100-2x^50-4x^25+3x

=x(x^99-2x^49-4x^24+3)

=x{x^49(x^50 - 1) - x^24(x^25+1) - 3(x^24 - 1)}

=x{x^49(x+1)(x^24-.....+1)(x^25-1) - x^24(x+1)(x^24-.....+1) +3(x+1)(x^2-x+1)(x^3-1)(x^6+1)(x^12+1)

=x(x+1)(x^98+.........................+3)

Thus,(x+1) is one of the factor