Question

graph the polynomial function f(x)=(x + 1)^2(x - 5)^2 using parts (a) through (e). (type a whole number.) (e) use the information to draw a complete graph of the function. choose the correct graph.

Explanation:

Step1: Find the x - intercepts

Set \(f(x)=(x + 1)^2(x - 5)^2=0\). By the zero - product property, \(x+1 = 0\) gives \(x=-1\) and \(x - 5=0\) gives \(x = 5\). The x - intercepts are \(x=-1\) and \(x = 5\). Since the exponents of the factors are 2 (even), the graph touches the x - axis at these points.

Step2: Find the y - intercept

Set \(x = 0\), then \(f(0)=(0 + 1)^2(0 - 5)^2=1\times25 = 25\). The y - intercept is \(y = 25\).

Step3: Determine the end - behavior

The leading term of the polynomial \(f(x)=(x + 1)^2(x - 5)^2=(x^{2}+2x + 1)(x^{2}-10x + 25)=x^{4}-8x^{3}-9x^{2}+50x + 25\). Since the leading coefficient is positive (the coefficient of \(x^{4}\) is 1) and the degree \(n = 4\) (even), as \(x\to\pm\infty\), \(y\to+\infty\).

Answer:

The graph that touches the x - axis at \(x=-1\) and \(x = 5\), has a y - intercept of 25 and goes to \(+\infty\) as \(x\to\pm\infty\) is the correct one. Without seeing the detailed features of each graph, based on the above - mentioned characteristics, we can analyze each option. But if we assume the standard behavior of a quartic polynomial with positive leading coefficient and the given x and y intercepts, we can make a choice. If graph A has these features (touches at \(x=-1\) and \(x = 5\), positive y - intercept, and correct end - behavior), then the answer is A. You need to check the details of each graph against the above - described properties.