What are the possible rational roots of #x^3-5x^2-4x+20=0# and then determine the rational roots?

1 Answer
Dec 12, 2016

Answer:

The possible rational roots are #+-1,+-2,+-4,+-5,+-10,+-20#. The rational roots are #x=-2, x=2, x=5#.

Explanation:

#color(blue)1x^3-5x^2-4x+color(red)(20)=0#

The possible rational roots are the factors of the constant #color(red)20# divided by the factors of the leading coefficient #color(blue)1#. The factors of the constant are called #p# and the factors of the leading coefficient are called #q#.

#p/q=(+-1,+-2,+-4,+-5,+-10,+-20)/(+-1)=#

#+-1,+-2,+-4,+-5,+-10,+-20#

The rational roots of this particular function can be found by factoring.

Factor by grouping.

First, group the first two terms and the second two terms.

#(x^3-5x^2)color(white)a+(-4x+20)=0#

Factor out a GCF from each group.

#x^2(x-5)-4(x-5)=0#

Regroup.

#(x^2-4)(x-5)=0#

Factor #x^2-4# as the difference of squares.

#(x+2)(x-2)(x-5)=0#

Set each factor equal to zero and solve.

#x+2=0color(white)(aa)x-2=0color(white)(aa)x-5=0#

#x=-2color(white)(aaa)x=2color(white)(aaa)x=5#