Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=x²(x-5)(x+3)(x-1)

Verified step by step guidance

1. Identify the given polynomial function: \(f(x) = x^2 (x - 5)(x + 3)(x - 1)\).
2. Since the polynomial is already factored, note the roots by setting each factor equal to zero: \(x^2 = 0\), \(x - 5 = 0\), \(x + 3 = 0\), and \(x - 1 = 0\).
3. Solve each equation to find the zeros of the function: \(x = 0\) (with multiplicity 2), \(x = 5\), \(x = -3\), and \(x = 1\).
4. Determine the behavior of the graph at each zero, especially noting that \(x=0\) has multiplicity 2, which means the graph touches the x-axis and turns around at this point, while the other zeros with multiplicity 1 cross the x-axis.
5. Choose test points in each interval determined by the zeros to find the sign of \(f(x)\) in those intervals, then sketch the graph accordingly, considering the end behavior based on the leading term when the polynomial is expanded.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding the degree and leading term helps predict the general shape and end behavior of the graph.

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of simpler polynomials or factors. This process reveals the roots or zeros of the function, which correspond to the x-intercepts on the graph, making it easier to plot the function accurately.

Graphing Polynomial Functions

Graphing involves plotting key points such as zeros, determining the multiplicity of roots to understand the behavior at intercepts, and analyzing end behavior based on the leading term. This helps visualize the shape and important features of the polynomial.