Question

Enter the correct answer in the box. Consider the graph of the sixth-degree polynomial function f. Replace the values b, c, and d to write function f.
f(x)=(x-b)(x-c)^2(x-d)^3

Answer 1

Answer: (x - 1)(x + 1)^2(x - 4)^3

Step-by-step explanation:

First, remember that:

in expressions like: (x - b)^n

b is the value of x where the graph intersects the x-axis.

n can represent:

n = 1, the line just goes through the x-axis

n = 2, the line may change the direction (an inflection point), and touch the x-axis in one point.

n = 3, the line may have two inflection points when it intersects the x-axis.

Then we have the expression:

(x-b)(x-c)^2(x-d)^3

b is in the linear part, the graph crosses the x-axis linearly in x = 1.

c is in the quadratic part, the graph crosses the x-axis with one point of inflection at x = -1.

d is in the cubic part, the graph crosses the x-axis with two inflections in x = 4.

Then we can writhe the polynomial as:

f(x) = (x-1)(x-(-1))^2(x-4)^3 = (x - 1)(x + 1)^2(x - 4)^3