Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=(3x-1)(x+2)²

Verified step by step guidance

1. Identify the given polynomial function: \(f(x) = (3x - 1)(x + 2)^2\). Since it is already factored, we can proceed to analyze it for graphing.
2. Determine the zeros of the function by setting each factor equal to zero: solve \$3x - 1 = 0\( and \)x + 2 = 0$ to find the x-intercepts.
3. Analyze the multiplicity of each zero: the factor \((3x - 1)\) has multiplicity 1, and \((x + 2)^2\) has multiplicity 2. This affects the behavior of the graph at these points.
4. Find the y-intercept by evaluating \(f(0)\): substitute \(x = 0\) into the function and simplify to get the y-coordinate where the graph crosses the y-axis.
5. Determine the end behavior of the polynomial by considering the degree and leading coefficient: since \((x + 2)^2\) is squared and \$3x - 1\( is linear, multiply their leading terms to find the overall degree and leading coefficient, which will guide how the graph behaves as \)x$ approaches positive and negative infinity.

Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to whole-number exponents and their coefficients. Understanding their general shape and behavior helps in graphing, especially recognizing how degree and leading coefficients affect the graph's end behavior.

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 are critical points where the graph intersects the x-axis.

Graphing Using Zeros and Multiplicities

The zeros of a polynomial correspond to x-intercepts on the graph. The multiplicity of each zero determines the graph's behavior at that point: odd multiplicities cross the x-axis, while even multiplicities touch and turn around.