Question

Use the Rational Zero Theorem to list all possible rational zeros for the given function. f(x) = 10x^5 + 8x^4 - 15x^3 +2x^2 - 2

Answer 1

Given:

The function is:

`f(x)=10x^5+8x^4-15x^3+2x^2-2`

To find:

All the possible rational zeros for the given function by using the Rational Zero Theorem.

Solution:

According to the rational root theorem, all the rational roots are of the form
`(p)/(q),q\\eq 0`, where p is a factor of constant term and q is a factor of leading coefficient.

We have,

`f(x)=10x^5+8x^4-15x^3+2x^2-2`

Here,

Constant term = -2

Leading coefficient = 10

Factors of -2 are ±1, ±2.

Factors of 10 are ±1, ±2, ±5, ±10.

Using the rational root theorem, all the possible rational roots are:

`x=\pm 1,\pm 2,\pm (1)/(2), \pm (1)/(5),\pm (2)/(5),\pm (1)/(10)`.

Therefore, all the possible rational roots of the given function are
`\pm 1,\pm 2,\pm (1)/(2), \pm (1)/(5),\pm (2)/(5),\pm (1)/(10)`.