Question

One factor of the function f(x) = x^3 − 9x^2 + 20x − 12 is (x − 6). Describe how to find the x-intercepts and the y-intercept of the graph of f(x) without using technology. Show your work and include all intercepts in your answer.

Answer 1

The Function is:

F(X) = X³ - 9X² + 20X -12

One of the roots is given, when in the statement it was said that X-6, so the root its +6......

Dividing the function by X - 6, we get the other 2 roots, because the function is composed by the multiplication of the roots in the form:

F(X) = aX³ + bX² + cX - d

(X - P).(X - Q).(X - R)   (where P, Q and R are the roots)

In this problem we already have X - 6, SO

(X - P).(X - Q).(X - 6)

(X³ - 9X² + 20X -12) ÷ (X - 6) = (X - P).(X - Q)

(X³ - 9X² + 20X -12) ÷ (X - 6) = X² - 3X + 2

X² - 3X + 2

Δ = 1

X' = 2 and X'' = 1

(X - P).(X - Q).(X - 6)  ⇔ (X - 2).(X - 1).(X - 6)

So

(X³ - 9X² + 20X -12) = (X - 2).(X - 1).(X - 6)

The roots are: 2, 1 and 6....  these are the points that the function crosses the x axis

F(X) = aX³ + bX² + cX - d

d = point where the function crosses the y axis

So 12 y-intercept of the graph of f(x)