Question

How do you find all the zeros of `f(x)=2x^3-13x^2+22x-8`?

Answer 1

Use the Rational Factor Theorem to identify rational zeros.

Explanation:

According to the Rational Factor Theorem
if `2x^3-13x^2+22x-8` has rational zeros
they must be in the set
`color(white)("XXX")+-("factors of 8")/("factors of 2")=+-({1,2,4,8})/({1,2})`
Evaluating the equation for each possibility:

identifies the zeros as `{color(red)(1/2),color(blue)(2),color(brown)(4)}`
which implies the factoring:
`color(white)("XXX")(color(red)(2x-1))(color(blue)(x-2))(color(brown)(x-4))`