Question

graph the polynomial function f(x)=(x + 1)^2(x - 5)^2 using parts (a) through (e). (a) determine the end - behavior of the graph of the function. the graph of f behaves like y = for large values of |x|.

Explanation:

Step1: Expand the polynomial

\[

$$\begin{align*} f(x)&=(x + 1)^2(x - 5)^2\\ &=(x^{2}+2x + 1)(x^{2}-10x + 25)\\ &=x^{2}(x^{2}-10x + 25)+2x(x^{2}-10x + 25)+1(x^{2}-10x + 25)\\ &=x^{4}-10x^{3}+25x^{2}+2x^{3}-20x^{2}+50x+x^{2}-10x + 25\\ &=x^{4}-8x^{3}+6x^{2}+40x + 25 \end{align*}$$

\]

Step2: Determine end - behavior

For a polynomial function \(y = a_nx^n+a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0\), the end - behavior is determined by the leading term \(a_nx^n\). Here, \(n = 4\) (even) and \(a_n=1>0\). So for large values of \(|x|\), the graph of \(f\) behaves like \(y = x^{4}\).

Answer:

\(x^{4}\)