Question

The Depressed Cubic: This problem explains how every polynomial of the form f(x) = ax^3 + bx^2 + cx + d can be transformed into a polynomial of the form g(t) = t^3 + pt + q a) Let f(x) = ax^3 + bx^2 + cx + d . Find an equation in terms of a, b for the x-value of the inflection point of f(x). Call that point r. (Hint: Use the second derivative) b) Explain what the substitution g(t) = f(x - r) does to the inflection point (use pictures to help with the explanation) c) Show that g(t) = f(x - r) has the form g(t) = x^3 + px + q and write p and q in terms of a, b, c and d. (g(t) is called the depressed cubic)

          The Depressed Cubic: This problem explains how every polynomial of the form
f(x) = ax^3 + bx^2 + cx + d can be transformed into a polynomial of the form
g(t) = t^3 + pt + q

a) Let f(x) = ax^3 + bx^2 + cx + d . Find an equation in terms of a, b for the x-value
of the inflection point of f(x). Call that point r. (Hint: Use the second derivative)

b) Explain what the substitution g(t) = f(x - r) does to the inflection point
(use pictures to help with the explanation)

c) Show that g(t) = f(x - r) has the form g(t) = x^3 + px + q and write p and q in
terms of a, b, c and d. (g(t) is called the depressed cubic)

The Depressed Cubic: This problem explains how every polynomial of the form
f(x) = ax^3 + bx^2 + cx + d can be transformed into a polynomial of the form
g(t) = t^3 + pt + q

a) Let f(x) = ax^3 + bx^2 + cx + d . Find an equation in terms of a, b for the x-value
of the inflection point of f(x). Call that point r. (Hint: Use the second derivative)

b) Explain what the substitution g(t) = f(x - r) does to the inflection point
(use pictures to help with the explanation)

c) Show that g(t) = f(x - r) has the form g(t) = x^3 + px + q and write p and q in
terms of a, b, c and d. (g(t) is called the depressed cubic)

Added by Marie H.

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