rational zeros

Question

Find all rational zeroes of the polynomial and write in complete factored form: g(x)=x^3-7x^2+14x-8

Answer 1

x^3-7x^2+14x-8=g(x)

try factors of 8 such as 1,2,4,8,-1,-2,-4,-8

2 is a zero

2^3-7*4+14*2-8=8-28+28-8=0

 

      x^2-5x+4

     __________________

x-2l x^3-7x^2+14x-8

      x^3-2x^2

     _________

            -5x^2+14x

            -5x^2+10x

           ___________

                         4x-8

                         4x-8

                         ____

                              0

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

(x-2)(x-4)(x-1)=0

x=2,4, and 1

Answer 2

g(x) = x³ - 7x² + 14x - 8

 

By the Rational Zero Theorem, the possible rational zeros are 

±1, ±2, ±4, ±8.

 

By trial and error, g(2) = 0, So, x - 2 is a factor of g(x).

 

Divide g(x) by x - 2:

 

2      1        -7       14          -8

                    2      -10           8

       1         -5         4            0

 

g(x) = (x - 2)(x² - 5x + 4)

         

         = (x - 2)(x - 4)(x - 1) 

 

Setting g(x) equal to zero, the roots of g(x) are 2, 4, and 1.