Question

graph the polynomial function f(x)=(x + 1)^2(x - 5)^2 using parts (a) through (e). (c) determine the zeros of the function and their multiplicity. use this information to determine whether or touches the x - axis at each x - intercept. the zero(s) of f is/are - 1,5. (simplify your answer. type an integer or a fraction. use a comma to separate answers as needed. t only once.) the lesser zero is a zero of multiplicity 2, so the graph of f touches the x - axis at x = - 1. the great of multiplicity 2, so the graph of f touches the x - axis at x = 5. (d) determine the maximum number of turning points on the graph of the function. (type a whole number.)

Explanation:

Step1: Recall the formula for the maximum - number of turning points

The maximum number of turning points of a polynomial function $y = f(x)$ of degree $n$ is given by $n - 1$.

Step2: Determine the degree of the polynomial function

Expand $f(x)=(x + 1)^2(x - 5)^2=(x^2+2x + 1)(x^2-10x + 25)$.
Using the distributive property (FOIL extended), we get $f(x)=x^4-10x^3+25x^2+2x^3-20x^2 + 50x+x^2-10x + 25=x^4-8x^3+6x^2+40x + 25$. The degree $n = 4$.

Step3: Calculate the maximum number of turning points

Substitute $n = 4$ into the formula $n - 1$. So, the maximum number of turning points is $4-1=3$.

Answer:

3