Polynomial Check 24A03F

1. **Problem Statement:** Determine if each given function is a polynomial function. If yes, write it in standard form, and identify the leading term, leading coefficient, and degree. 2. **Recall:** A polynomial function is a function of the form $$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$ where $n$ is a non-negative integer and all coefficients $a_i$ are real numbers. 3. **Part (a):** $v_0 t - \frac{1}{2} t^2$ - This is a sum of terms with variables raised to whole number powers: $t^1$ and $t^2$. - The function can be written as $$-\frac{1}{2} t^2 + v_0 t$$ in standard form (descending powers). - Leading term: $-\frac{1}{2} t^2$ - Leading coefficient: $-\frac{1}{2}$ - Degree: $2$ - Conclusion: This is a polynomial function. 4. **Part (b):** $f(x) = \frac{5}{3} x^2 - \sqrt{7} x^2 + 8 x^3 - \frac{1}{2} + \ldots$ - The terms shown are $8 x^3$, $\frac{5}{3} x^2$, $-\sqrt{7} x^2$, and $-\frac{1}{2}$. - Combine like terms for $x^2$: $$\frac{5}{3} x^2 - \sqrt{7} x^2 = \left(\frac{5}{3} - \sqrt{7}\right) x^2$$ - The function in standard form (descending powers) is: $$f(x) = 8 x^3 + \left(\frac{5}{3} - \sqrt{7}\right) x^2 - \frac{1}{2} + \ldots$$ - Since the ellipsis ($\ldots$) is incomplete, we cannot confirm if the function contains only polynomial terms. - If the omitted terms are polynomial terms (powers of $x$ with non-negative integers), then $f(x)$ is a polynomial. - Leading term: $8 x^3$ - Leading coefficient: $8$ - Degree: $3$ 5. **Summary:** - (a) is a polynomial function of degree 2 with leading term $-\frac{1}{2} t^2$ and leading coefficient $-\frac{1}{2}$. - (b) is a polynomial function of degree 3 with leading term $8 x^3$ and leading coefficient $8$ if the omitted terms are polynomial terms; otherwise, more information is needed.