To find the factors of the polynomial function $f(x) = 2x^3 + x^2 - 8x - 4$, let's analyze it step-by-step. We need to determine which of the options $(x+4)$, $(x-2)$, $(2x+1)$, $(x+2)$, and $(x-4)$ are factors of this polynomial.

Steps:

1. Identify Possible Rational Roots: Since this is a cubic polynomial, we can use the Rational Root Theorem to find possible roots. This theorem suggests testing the factors of the constant term $-4$ over the factors of the leading coefficient $2$.

   Possible rational roots are: $\pm 1, \pm 2, \pm 4, \pm \frac{1}{2}$.

2. Test These Roots by Substitution: We can substitute these values into $f(x)$ to check if they yield $f(x) = 0$, which would indicate a root.

   • Testing $x = 2$: $f(2) = 2(2)^3 + (2)^2 - 8(2) - 4 = 2 \cdot 8 + 4 - 16 - 4 = 16 + 4 - 16 - 4 = 0$ Since $f(2) = 0$, $x = 2$ is a root, meaning $(x - 2)$ is a factor.

   • Testing $x = -2$: $f(-2) = 2(-2)^3 + (-2)^2 - 8(-2) - 4 = 2(-8) + 4 + 16 - 4 = -16 + 4 + 16 - 4 = 0$ Since $f(-2) = 0$, $x = -2$ is a root, meaning $(x + 2)$ is a factor.

   • Testing $x = -\frac{1}{2}$:

     = 2\left(-\frac{1}{8}\right) + \frac{1}{4} + 4 - 4 = -\frac{1}{4} + \frac{1}{4} + 4 - 4 = 0$$ Since $$f\left(-\frac{1}{2}\right) = 0$$, $$x = -\frac{1}{2}$$ is a root, meaning $$(2x + 1)$$ is a factor.

Conclusion:

The factors of $f(x) = 2x^3 + x^2 - 8x - 4$ are:

• $(x - 2)$
• $(x + 2)$
• $(2x + 1)$

So, the correct options to select are:

• $(x-2)$
• $(x+2)$
• $(2x+1)$

Would you like more details on each calculation step or have any questions?

Related Questions:

1. What is the Rational Root Theorem and how does it work?
2. How can we factor a cubic polynomial using synthetic division?
3. What other methods can be used to find polynomial factors?
4. Why is testing roots helpful in factoring polynomials?
5. How would the factors change if the polynomial had an additional term?

Tip:

Always verify possible roots by substituting them back into the polynomial to confirm they yield zero; this ensures accuracy in determining factors.