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

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