Math Snap

STEP 1

1. We are given the polynomial function $f(x) = 4x^4 + 8x^3 - 3x^2 - 7x - 2$.
2. We need to find the roots of the polynomial, i.e., the values of $x$ for which $f(x) = 0$.
3. We have multiple choice options to evaluate.

STEP 2

1. Use the Rational Root Theorem to identify possible rational roots.
2. Test the possible roots using synthetic division or direct substitution.
3. Determine which option contains the correct roots.

STEP 3

Apply the Rational Root Theorem. The possible rational roots are the factors of the constant term $-2$ divided by the factors of the leading coefficient $4$.
Possible rational roots: $\pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{4}$.

STEP 4

Test each possible root by substituting into $f(x)$ to see if it equals zero.
Test $x = -2$:
$$f(-2) = 4(-2)^4 + 8(-2)^3 - 3(-2)^2 - 7(-2) - 2$$ $$f(-2) = 4(16) + 8(-8) - 3(4) + 14 - 2$$ $$f(-2) = 64 - 64 - 12 + 14 - 2$$ $$f(-2) = 0$$ $x = -2$ is a root.

STEP 5

Test $x = 1$:
$$f(1) = 4(1)^4 + 8(1)^3 - 3(1)^2 - 7(1) - 2$$ $$f(1) = 4 + 8 - 3 - 7 - 2$$ $$f(1) = 0$$ $x = 1$ is a root.

STEP 6

Test $x = -\frac{1}{2}$:
$$f\left(-\frac{1}{2}\right) = 4\left(-\frac{1}{2}\right)^4 + 8\left(-\frac{1}{2}\right)^3 - 3\left(-\frac{1}{2}\right)^2 - 7\left(-\frac{1}{2}\right) - 2$$ $$f\left(-\frac{1}{2}\right) = 4\left(\frac{1}{16}\right) + 8\left(-\frac{1}{8}\right) - 3\left(\frac{1}{4}\right) + \frac{7}{2} - 2$$ $$f\left(-\frac{1}{2}\right) = \frac{1}{4} - 1 - \frac{3}{4} + \frac{7}{2} - 2$$ $$f\left(-\frac{1}{2}\right) = 0$$ $x = -\frac{1}{2}$ is a root.

Based on the tests, the roots are $x = -2$, $x = 1$, and $x = -\frac{1}{2}$.
The correct option is:
B. $x = -2$ or $x = 1$ or $x = -\frac{1}{2}$
The correct answer is option B.