Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4.
$\text{ƒ}\left(x\right)=x^3+x^2-36x-36$

Verified step by step guidance

1. Start by writing down the polynomial function: \(f(x) = x^3 + x^2 - 36x - 36\).
2. Look for common factors or use factoring by grouping. Group the terms as \((x^3 + x^2) + (-36x - 36)\).
3. Factor out the greatest common factor (GCF) from each group: \(x^2(x + 1) - 36(x + 1)\).
4. Notice that \((x + 1)\) is a common binomial factor, so factor it out: \((x + 1)(x^2 - 36)\).
5. Recognize that \(x^2 - 36\) is a difference of squares, which factors as \((x - 6)(x + 6)\), so the fully factored form is \((x + 1)(x - 6)(x + 6)\).

Key Concepts

Polynomial Functions

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

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of simpler polynomials or factors. This process helps identify the roots or zeros of the function, which are critical points where the graph intersects the x-axis.

Graphing Polynomial Functions

Graphing a polynomial requires plotting its zeros, analyzing end behavior, and identifying turning points. Factoring first simplifies finding zeros, and understanding multiplicity of roots helps determine whether the graph crosses or touches the x-axis at those points.