1. Let's start by stating the problem: We want to understand what a polynomial function is and how to work with it. 2. A polynomial function is a function of the form $$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$ where $a_n, a_{n-1}, \ldots, a_0$ are constants called coefficients, and $n$ is a non-negative integer called the degree of the polynomial. 3. Important rules: - The degree $n$ tells us the highest power of $x$ in the polynomial. - Coefficients can be any real numbers. - Polynomial functions are continuous and smooth. 4. Example: Consider the polynomial $$f(x) = 2x^3 - 4x^2 + 3x - 5$$ - Here, the degree is 3 because the highest power of $x$ is 3. - The coefficients are 2, -4, 3, and -5. 5. To evaluate a polynomial at a specific value, substitute the value of $x$ and simplify. For example, to find $f(2)$: $$f(2) = 2(2)^3 - 4(2)^2 + 3(2) - 5 = 2(8) - 4(4) + 6 - 5 = 16 - 16 + 6 - 5 = 1$$ 6. Polynomials can be added, subtracted, multiplied, and divided (except by zero). When multiplying, use the distributive property and combine like terms. 7. Factoring polynomials involves expressing them as a product of simpler polynomials, which helps in solving equations. 8. Summary: Polynomial functions are expressions involving powers of $x$ with constant coefficients. They are fundamental in algebra and appear in many areas of math and science.