How do you find the roots of #x^3-5x^2-x+5=0#?

1 Answer
Oct 13, 2016

Answer:

Use factoring by grouping to get #x=-1, x=1, x=5#.

Explanation:

#x^3-5x^2-x+5=0#

Use a method called "factoring by grouping", which only works when the polynomial has four terms.

#color(red)(x^2)(x-5)-x+5=0color(white)(aaa)#Factor out an #color(red)(x^2)# from the first two #color(white)(aaaaaaaaaaaaaaaaaaaaa)#terms.

#color(red)(x^2)(x-5)color(blue)(-1)(x-5)=0color(white)(aaa)#Factor out a #color(blue)(-1)# from the 2nd two #color(white)(aaaaaaaaaaaaaaaaaaaaaa)# terms.

#(color(red)(x^2)color(blue)(-1))(x-5)=0color(white)(aaa)#Regroup

#(x+1)(x-1)(x-5)=0color(white)(aaa)#Factor the #(x^2-1)# term using the #color(white)(aaaaaaaaaaaaaaaaaaaaaaaa)# difference of squares formula
#color(white)(aaaaaaaaaaaaaaaaaaaaaaaa)(a^2-b^2)=(a+b)(a-b)#

Set each factor equal to zero and solve.

#x+1=0, x-1=0, x-5=0#

#x=-1, x=1, x=5#