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

Let's analyze each function to determine whether it is a polynomial. Recall that a **polynomial** is an expression consisting of variables (also called indeterminates) and coefficients, combined using only addition, subtraction, multiplication, and non-negative integer exponents of variables. If a function is a polynomial, we'll rewrite it in **standard form**, which arranges the terms in descending order of their exponents. ### a) \( f(x) = -3x + 5x^{3} - 6x^{2} + 2 \) **Analysis:** - All exponents of \( x \) are non-negative integers (3, 2, 1, 0). - No variables have exponents as fractions, negatives, or are under radicals. **Conclusion:** - **Polynomial:** Yes - **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 rewritten as \( -5x^{-1} \), which has a negative exponent. - Negative exponents are not allowed in polynomials. **Conclusion:** - **Polynomial:** No ### c) \( f(x) = 9x^{4} + 8x^{3} - 6x^{-2} + 2x \) **Analysis:** - The term \( -6x^{-2} \) has a negative exponent. - Negative exponents are not allowed in polynomials. **Conclusion:** - **Polynomial:** No ### d) \( g(x) = \sqrt{3} - 12x + 13x^{2} \) **Analysis:** - All exponents of \( x \) are non-negative integers (2, 1, 0). - \( \sqrt{3} \) is a constant term and does not involve any variables. **Conclusion:** - **Polynomial:** Yes - **Standard Form:** \( g(x) = 13x^{2} - 12x + \sqrt{3} \) ### e) \( p(x) = \frac{1}{2}x^{2} + 3x - 4x^{3} + 6x^{1} - 1 \) **Analysis:** - All exponents of \( x \) are non-negative integers (3, 2, 1, 0). - Combine like terms if necessary. **Simplification:** - Combine \( 3x \) and \( 6x \): \( 3x + 6x = 9x \) **Conclusion:** - **Polynomial:** Yes - **Standard Form:** \( p(x) = -4x^{3} + \frac{1}{2}x^{2} + 9x - 1 \) ### f) \( h(x) = \frac{5}{3}x^{2} - \sqrt{7}x^{4} + 8x^{3} - \frac{1}{2} + x \) **Analysis:** - All exponents of \( x \) are non-negative integers (4, 3, 2, 1, 0). - Coefficients can be constants, including radicals or fractions. **Conclusion:** - **Polynomial:** Yes - **Standard Form:** \( h(x) = -\sqrt{7}x^{4} + 8x^{3} + \frac{5}{3}x^{2} + x - \frac{1}{2} \) --- **Summary:** | Part | Is Polynomial? | Standard Form | |------|----------------|---------------| | a) | Yes | \( 5x^{3} - 6x^{2} - 3x + 2 \) | | b) | No | — | | c) | No | — | | d) | Yes | \( 13x^{2} - 12x + \sqrt{3} \) | | e) | Yes | \( -4x^{3} + \frac{1}{2}x^{2} + 9x - 1 \) | | f) | Yes | \( -\sqrt{7}x^{4} + 8x^{3} + \frac{5}{3}x^{2} + x - \frac{1}{2} \) | If you have any further questions or need additional explanations, feel free to ask!