1. Identify whether the following functions are polynomial or not, and if polynomial, find degree, leading coefficient, and constant term. **a.** $$f(x) = 5x^2 - \frac{2}{3}x^3 - \frac{1}{3}x - \frac{4x^4 + 9x^3 - 2x + 9}{3}$$ Step 1: Simplify the expression inside the fraction: $$\frac{4x^4 + 9x^3 - 2x + 9}{3} = \frac{4x^4}{3} + 3x^3 - \frac{2x}{3} + 3$$ Step 2: Rewrite entire function: $$f(x) = 5x^2 - \frac{2}{3}x^3 - \frac{1}{3}x - \left(\frac{4x^4}{3} + 3x^3 - \frac{2x}{3} + 3\right)$$ Step 3: Distribute the minus sign: $$f(x) = 5x^2 - \frac{2}{3}x^3 - \frac{1}{3}x - \frac{4x^4}{3} - 3x^3 + \frac{2x}{3} - 3$$ Step 4: Combine like terms: - For $x^4$: $-\frac{4}{3}x^4$ - For $x^3$: $-\frac{2}{3}x^3 - 3x^3 = -\frac{2}{3}x^3 - \frac{9}{3}x^3 = -\frac{11}{3}x^3$ - For $x$: $-\frac{1}{3}x + \frac{2}{3}x = \frac{1}{3}x$ - For constant: $-3$ Step 5: Final simplified polynomial: $$f(x) = -\frac{4}{3}x^4 - \frac{11}{3}x^3 + 5x^2 + \frac{1}{3}x - 3$$ - Degree: 4 - Leading coefficient: $-\frac{4}{3}$ - Constant term: $-3$ **b.** $$f(x) = 2\left(\frac{1}{3}\right) + \frac{3}{2} - \frac{1}{6}$$ Step 1: Calculate constants: $$2 \times \frac{1}{3} = \frac{2}{3}$$ Step 2: Sum all constants: $$\frac{2}{3} + \frac{3}{2} - \frac{1}{6} = \frac{4}{6} + \frac{9}{6} - \frac{1}{6} = \frac{12}{6} = 2$$ This is a constant function, which is a polynomial of degree 0. - Degree: 0 - Leading coefficient: 2 - Constant term: 2 **c.** $$f(x) = 3(x^2)^3 - 4(x^2)^2 + 2(x+1)^2 + 5$$ Step 1: Simplify powers: $$(x^2)^3 = x^{6}, \quad (x^2)^2 = x^{4}$$ Step 2: Expand $(x+1)^2 = x^2 + 2x + 1$ Step 3: Substitute: $$f(x) = 3x^6 - 4x^4 + 2(x^2 + 2x + 1) + 5 = 3x^6 - 4x^4 + 2x^2 + 4x + 2 + 5$$ Step 4: Combine constants: $$f(x) = 3x^6 - 4x^4 + 2x^2 + 4x + 7$$ - Degree: 6 - Leading coefficient: 3 - Constant term: 7 **d.** $$f(x) = -2(\sqrt{x})^3 + 5\sqrt{x} - 10$$ Step 1: Rewrite powers: $$(\sqrt{x})^3 = (x^{1/2})^3 = x^{3/2}$$ Step 2: Expression becomes: $$f(x) = -2x^{3/2} + 5x^{1/2} - 10$$ Since exponents are not whole numbers, this is not a polynomial. **e.** $$f(x) = 3\pi^2 + 4$$ This is a constant function (number), so polynomial degree 0. - Degree: 0 - Leading coefficient: $3\pi^2 + 4$ - Constant term: $3\pi^2 + 4$ Summary for question 1: - a: Polynomial, degree 4, leading coefficient $-\frac{4}{3}$, constant $-3$ - b: Polynomial, degree 0, leading coefficient 2, constant 2 - c: Polynomial, degree 6, leading coefficient 3, constant 7 - d: Not polynomial - e: Polynomial, degree 0, leading coefficient $3\pi^2 + 4$, constant $3\pi^2 + 4$