Stay whether the function is a polynomial or not. If it is, write it in standard form. \( \begin{array}{ll}\text { a) } f(x) \frac{\text { s? }^{\text {my }}}{}-3 x+5 x^{3}-6 x^{2}+2 & \text { b) } h(x)=3 x^{4}+2 x-\frac{5}{x}+9 x^{3}-7 \\ \text { c) } f(x)=9 x^{4}+8 x^{3}-6 x^{-2}+2 x & \text { d) } g(x)=\sqrt{3}-12 x+13 x^{2} \\ \text { e) } p(x)=\frac{1}{2} x^{2}+3 x-4 x^{3}+6 x^{4}-1 & \text { f) } h(x)=\frac{5}{3} x^{2}-\sqrt{7} x^{4}+8 x^{3}-\frac{1}{2}\end{array} \)

Let's analyze each function to determine whether it is a polynomial. If it is, we'll rewrite it in standard form, which arranges the terms in descending order of their degrees. ### a) \( f(x) = -3x + 5x^3 - 6x^2 + 2 \) - **Analysis:** All exponents are non-negative integers (3, 2, 1, and 0). - **Conclusion:** **Polynomial** - **Standard Form:** \( f(x) = 5x^3 - 6x^2 - 3x + 2 \) --- ### b) \( h(x) = 3x^4 + 2x - \frac{5}{x} + 9x^3 - 7 \) - **Analysis:** The term \( \frac{5}{x} \) can be written as \( -5x^{-1} \), which has a negative exponent. - **Conclusion:** **Not a polynomial** --- ### c) \( f(x) = 9x^4 + 8x^3 - 6x^{-2} + 2x \) - **Analysis:** The term \( -6x^{-2} \) has a negative exponent. - **Conclusion:** **Not a polynomial** --- ### d) \( g(x) = \sqrt{3} - 12x + 13x^2 \) - **Analysis:** All exponents are non-negative integers (2, 1, and 0). The coefficient \( \sqrt{3} \) is a constant. - **Conclusion:** **Polynomial** - **Standard Form:** \( g(x) = 13x^2 - 12x + \sqrt{3} \) --- ### e) \( p(x) = \frac{1}{2}x^2 + 3x - 4x^3 + 6x^4 - 1 \) - **Analysis:** All exponents are non-negative integers (4, 3, 2, 1, and 0). - **Conclusion:** **Polynomial** - **Standard Form:** \( p(x) = 6x^4 - 4x^3 + \frac{1}{2}x^2 + 3x - 1 \) --- ### f) \( h(x) = \frac{5}{3}x^2 - \sqrt{7}x^4 + 8x^3 - \frac{1}{2} \) - **Analysis:** All exponents are non-negative integers (4, 3, 2, and 0). The coefficients \( \frac{5}{3} \) and \( \sqrt{7} \) are constants. - **Conclusion:** **Polynomial** - **Standard Form:** \( h(x) = -\sqrt{7}x^4 + 8x^3 + \frac{5}{3}x^2 - \frac{1}{2} \) --- ### Summary - **Polynomials:** - **a)** \( f(x) = 5x^3 - 6x^2 - 3x + 2 \) - **d)** \( g(x) = 13x^2 - 12x + \sqrt{3} \) - **e)** \( p(x) = 6x^4 - 4x^3 + \frac{1}{2}x^2 + 3x - 1 \) - **f)** \( h(x) = -\sqrt{7}x^4 + 8x^3 + \frac{5}{3}x^2 - \frac{1}{2} \) - **Not Polynomials:** - **b)** \( h(x) = 3x^4 + 2x - \frac{5}{x} + 9x^3 - 7 \) - **c)** \( f(x) = 9x^4 + 8x^3 - 6x^{-2} + 2x \)