What are all the possible rational zeros for #f(x)=5x^3+x^2-5x-1# and how do you find all zeros?

1 Answer
Sep 24, 2016

Answer:

The possible rational zeros are #-1, -1/5, 1/5, 5# and the zeros are #-1, -1/5, 1#

Explanation:

#f(x)=color(red)5x^3+x^2-5x-color(blue)1#

To find all the possible rational zeros, divide all the factors #p# of the constant term by all the factors #q# of the leading coefficient. The list of possible rational zeros is given by #p/q#.

The constant term #=color(blue)1# and the leading coefficient #=color(red)5#.

The factors #p# of the constant term #color(blue)1# are #+-1#.
The factors #q# of the leading coefficient #color(red)5# are #+-1# and #+-5#

#p/q#=#frac{+-1}{+-1,+-5}=1, -1, 1/5, -1/5#

The possible rational zeros are #-1, -1/5,1/5,1#

To find all the zeros, factor the polynomial and set each factor equal to zero.

#f(x)=5x^3+x^2-5x-1#

Factor by grouping.
#(5x^3+x^2)-(5x+1)#
#x^2(5x+1)-1(5x+1)#
#(x^2-1)(5x+1)#

Use the difference of squares to factor #x^2-1 #
#(x+1)(x-1)(5x+1)#

Set each factor equal to zero and solve.
#x+1=0color(white)(aaa)x-1=0color(white)(aaa)5x+1=0#

#x=-1color(white)(aaa)x=1color(white)(aaa)x=-1/5#