Use the rational root theorem to list all possible rational roots for the equation x³ - x² - x - 3 = 0
-3, -1, 1, 3
1, 3
-3, 3
No roots

Solution:

Using the rational root theorem, which states that, if the polynomial f(x) = a_nxⁿ + a_{n - 1}x^{n - 1} +...+ a₁x + a₀ has integer coefficients, then every rational zero of f(x) has the form p/q where p is a factor of the constant term a0 and q is a factor of the leading coefficient a_n.

The given equation is

x³ - x² - x - 3 = 0

We know that ±1 are ±3 factors of 3

f(x) = x³ - x² - x - 3 = 0

f(1) = 1³ - 1² - 1 - 3

f(1) = 1 - 1 - 1 - 3

f(1) = 1 - 5

f(1) = -4

f(-1) = (-1)³ - (-1)² - (-1) - 3

f(-1) = -1 - 1 + 1 - 3

f(-1) = -1 - 3

f(-1) = -4

f(3) = 3³ - 3² - 3 - 3

f(3) = 27 - 9 - 3 - 3

f(3) = 27 - 15

f(3) = 12

f(-3) = (-3)³ - (-3)² - (-3)- 3

f(-3) = -27 - 9 + 3 - 3

f(-3) = -39 + 3

f(-3) = -36

Therefore, there are no roots.

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Use the rational root theorem to list all possible rational roots for the equation x³ - x² - x - 3 = 0

Summary:

Using the rational root theorem, there are no roots for the equation x³ - x² - x - 3 = 0 are 1, -1, 3 and -3.