1. **State the problem:** Simplify the function $$y(x) = \frac{12x^6 - 4x^4 + 3x^2 - 1}{8x^4}$$ and understand its behavior. 2. **Formula and rules:** When dividing a polynomial by a monomial, divide each term in the numerator by the denominator separately. 3. **Divide each term:** $$y = \frac{12x^6}{8x^4} - \frac{4x^4}{8x^4} + \frac{3x^2}{8x^4} - \frac{1}{8x^4}$$ 4. **Simplify each term:** - For $$\frac{12x^6}{8x^4}$$, divide coefficients and subtract exponents: $$\frac{12}{8} = \frac{3}{2}$$ and $$x^{6-4} = x^2$$, so $$\frac{12x^6}{8x^4} = \frac{3}{2}x^2$$ - For $$\frac{4x^4}{8x^4}$$, divide coefficients and cancel variables: $$\frac{4}{8} = \frac{1}{2}$$ and $$\frac{x^4}{x^4} = 1$$, so $$\frac{4x^4}{8x^4} = \frac{1}{2}$$ - For $$\frac{3x^2}{8x^4}$$, divide coefficients and subtract exponents: $$\frac{3}{8}$$ and $$x^{2-4} = x^{-2}$$, so $$\frac{3x^2}{8x^4} = \frac{3}{8}x^{-2}$$ - For $$\frac{1}{8x^4}$$, it remains as is. 5. **Write the simplified expression:** $$y = \frac{3}{2}x^2 - \frac{1}{2} + \frac{3}{8}x^{-2} - \frac{1}{8x^4}$$ 6. **Interpretation:** The function is a rational expression simplified into a sum of terms with positive and negative powers of $$x$$. Final answer: $$y = \frac{3}{2}x^2 - \frac{1}{2} + \frac{3}{8}x^{-2} - \frac{1}{8x^4}$$