1. **Problem Statement:** Simplify and analyze the algebraic expression $$5x^3 - 7x^2 + 14 + 2x^3 - \frac{3x^4}{4}$$ and answer the following questions. 2. **Rewrite the expression in descending powers of $x$: ** Combine like terms and order by powers of $x$ from highest to lowest. 3. **Degree of the expression:** The degree is the highest power of $x$ in the expression. 4. **Smallest coefficient:** Identify the term with the smallest numerical coefficient. 5. **Constant term:** The term without $x$. 6. **Evaluate the expression at $x = -1$: ** Substitute $x = -1$ and simplify. --- **Step 1: Combine like terms and rewrite in descending powers:** Original expression: $$5x^3 - 7x^2 + 14 + 2x^3 - \frac{3x^4}{4}$$ Group like terms: $$- \frac{3x^4}{4} + (5x^3 + 2x^3) - 7x^2 + 14$$ Simplify: $$- \frac{3x^4}{4} + 7x^3 - 7x^2 + 14$$ **Answer 5.1.1:** $$- \frac{3x^4}{4} + 7x^3 - 7x^2 + 14$$ --- **Step 2: Degree of the expression** The highest power of $x$ is 4 (from $- \frac{3x^4}{4}$). **Answer 5.1.2:** 4 --- **Step 3: Smallest coefficient** Coefficients are: - $- \frac{3}{4} = -0.75$ (for $x^4$ term) - $7$ (for $x^3$ term) - $-7$ (for $x^2$ term) - $14$ (constant term) The smallest coefficient is $-7$. **Answer 5.1.3:** $-7$ --- **Step 4: Constant term** The constant term is $14$. **Answer 5.1.4:** 14 --- **Step 5: Evaluate the expression at $x = -1$** Substitute $x = -1$: $$- \frac{3(-1)^4}{4} + 7(-1)^3 - 7(-1)^2 + 14$$ Calculate powers: $$- \frac{3(1)}{4} + 7(-1) - 7(1) + 14 = - \frac{3}{4} - 7 - 7 + 14$$ Simplify: $$-0.75 - 7 - 7 + 14 = (-0.75 - 7 - 7) + 14 = -14.75 + 14 = -0.75$$ **Answer 5.1.5:** $-0.75$