Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=(4x+3)(x+2)²

Verified step by step guidance

1. Identify the given polynomial function: \(f(x) = (4x + 3)(x + 2)^2\). Notice that it is already expressed in factored form, so no further factoring is needed.
2. Recognize that the function is a product of two factors: a linear factor \((4x + 3)\) and a squared binomial factor \((x + 2)^2\). This means the graph will have zeros where each factor equals zero.
3. Find the zeros of the function by setting each factor equal to zero: solve \(4x + 3 = 0\) and \(x + 2 = 0\). The zero from \((x + 2)^2\) has multiplicity 2, which affects the shape of the graph at that zero.
4. Determine the behavior of the graph at each zero: at \(x = -\frac{3}{4}\) (from \(4x + 3 = 0\)), the graph crosses the x-axis because the factor is linear (multiplicity 1). At \(x = -2\) (from \((x + 2)^2\)), the graph touches the x-axis and turns around because the zero has even multiplicity (2).
5. Choose test points in the intervals determined by the zeros to find the sign of \(f(x)\) in each region, then plot the zeros and use the end behavior of the polynomial (leading term behavior) to sketch the graph accordingly.

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 general shape and behavior of polynomial graphs helps in sketching their curves accurately.

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 Factored Form

When a polynomial is in factored form, each factor corresponds to a root of the function. The multiplicity of each root affects the graph's behavior at that point, such as crossing or touching the x-axis, which guides accurate graphing.