Question

Use the rational root theorem to list all possible rational roots for the equation. X^3+2x-9=0.

Answer 1

The Rational Root Theorem allows us to find possible rational roots by dividing the factors of the constant term by the factors of the leading coefficient. For X^3+2x-9=0, this gives us possible roots of ±1, ±3, and ±9.

Step-by-step explanation:

The rational root theorem is a useful tool for identifying possible rational roots of polynomial equations. When applying it to the equation X^3+2x-9=0, the first step is to identify the factors of the constant term and the leading coefficient. For this equation, the constant term is -9 and its factors are ±1, ±3, ±9. The leading coefficient is 1, with the factors being just ±1. To list all possible rational roots, you take the factors of the constant term and divide them by the factors of the leading coefficient.

In this case, since the leading coefficient is 1, the possible rational roots are just the factors of the constant term, which are:

• ±1
• ±3
• ±9
• ±1/1
• ±3/1
• ±9/1

Therefore, the complete list of possible rational roots for the equation X^3+2x-9=0 is ±1, ±3, and ±9.

Answer 2

Rational root theorem:

Given a polynomial
`a_nx^n+a_(n-1)x^(n-1)+...+a_0`, the rational roots are +-
`(r_0)/(r_n)` where
`r_0` = factor of the constant
`a_0` and
`r_n` = factors of the leading coefficient
`a_n`.

So, to find the possible rational roots, we list all the factors of the constant and the leading coefficients and then set up the ratios.
In this case, the constant is
`a_0=–9` and the leading coefficient is
`a_n=1`.
Factors of –9: +-1, +-3, +-9
Factors of 1: +-1

Thus, the possible rational roots are +-1/1, +-3/1, +-9/1
or +-1, +-3, +-9.

Answer: +-1, +-3, +-9