Use remainder theorem

Question

Use the remainder theorem to find the remainder when f(x) is divided by x-1.  Then use the factor theorem to determine whether x-1 is a factor of f(x)=4x^4-5x^3-3x+4

Answer 1

f(x) = 4x^4-5x^3-3x+4

Recall remainder therom

Dividend = divisor*quotient+remainder

x-1)      4x^4-5x^3-3x+4          (4x^3-x^2-x-4

            4x^4-4x^3

           (-)     (+)   

         __________

                   -x^3+0*x^2-3x

                   -x^3+x^2

                  (+)   (-)

                  ___________

                     -x^2-3x+4

                    -x^2+x

                   (+)   (-)

                 _____________

                        -4x+4

                         -4x+4

                         (+) (-)

              ________________

                            0

Remainder is 0. quotient is 4x^3-x^2-x-4.

  (4x^3-x^2-x-4)(x-1)+0

=4x^4-4x^3-x^3+x^2-x^2+x-4x+4

= 4x^4-5x^3-3x+4

= f(x)

Factor therom

when f(c) = 0 then x-c is a factor of polynomial.

Here f(1) = 4*1-5*1-3*1+4

= 4-5-3+4 = 0

(x-1) is a factor of f(x).