Questions: Check all the possible rational roots for the function f(x)=3x^3+2x^2+3x+6 Select one or more: a. -1 / 2 b. -4 c. -1 / 3 d. -3 / 2 e. 6 f. 0 g. -2 / 3 h. 3

Solution Steps

To determine the possible rational roots of the polynomial \( f(x) = 3x^3 + 2x^2 + 3x + 6 \), we can use the Rational Root Theorem. This theorem states that any possible rational root, expressed as a fraction \( \frac{p}{q} \), must have \( p \) as a factor of the constant term (6) and \( q \) as a factor of the leading coefficient (3). We will then check each of the given options to see if they satisfy the polynomial equation \( f(x) = 0 \).

We are given the polynomial function: \[ f(x) = 3x^3 + 2x^2 + 3x + 6 \]

The given options for possible rational roots are: \[ -0.5, -4, -0.3333, -1.5, 6, 0, -0.6667, 3 \]

We need to check each option to see if it satisfies \( f(x) = 0 \).

1. For \( x = -0.5 \): \[ f(-0.5) = 3(-0.5)^3 + 2(-0.5)^2 + 3(-0.5) + 6 = -0.375 + 0.5 - 1.5 + 6 = 4.625 \neq 0 \]

2. For \( x = -4 \): \[ f(-4) = 3(-4)^3 + 2(-4)^2 + 3(-4) + 6 = -192 + 32 - 12 + 6 = -166 \neq 0 \]

3. For \( x = -0.3333 \): \[ f(-0.3333) \approx 3(-0.3333)^3 + 2(-0.3333)^2 + 3(-0.3333) + 6 \approx -0.0370 + 0.2222 - 0.9999 + 6 \approx 5.1853 \neq 0 \]

4. For \( x = -1.5 \): \[ f(-1.5) = 3(-1.5)^3 + 2(-1.5)^2 + 3(-1.5) + 6 = -10.125 + 4.5 - 4.5 + 6 = -4.125 \neq 0 \]

5. For \( x = 6 \): \[ f(6) = 3(6)^3 + 2(6)^2 + 3(6) + 6 = 648 + 72 + 18 + 6 = 744 \neq 0 \]

6. For \( x = 0 \): \[ f(0) = 3(0)^3 + 2(0)^2 + 3(0) + 6 = 6 \neq 0 \]

7. For \( x = -0.6667 \): \[ f(-0.6667) \approx 3(-0.6667)^3 + 2(-0.6667)^2 + 3(-0.6667) + 6 \approx -0.2963 + 0.4444 - 2.0001 + 6 \approx 4.1480 \neq 0 \]

8. For \( x = 3 \): \[ f(3) = 3(3)^3 + 2(3)^2 + 3(3) + 6 = 81 + 18 + 9 + 6 = 114 \neq 0 \]

Final Answer