One factor of the function f(x) = x3 − 9x2 + 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.

Step-by-step explanation:

Easy first: the y-intercept is the value of the constant in the function. That is the value of the function when x=0: -12.

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The x-intercepts of the function are where it crosses the x-axis. These are also called zeros or roots of the function. Each root (p) corresponds to a factor (x -p).

To find the other roots of the function, we need to factor out the given root. It is convenient to use synthetic division for the purpose. (In the attached, the calculation is done literally on the back of an envelope.) This results in a quadratic factor which can be further factored to find its zeros.

The synthetic division tells us the factorization is ...

  f(x) = (x -6)(x^2 -3x +2)

  f(x) = (x -6)(x -2)(x -1)

Then the x-intercepts are the values of x that make these factors zero:

  x = 1, 2, 6 and y = 0

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In general, you want to find linear or quadratic factors of the function so that you can use the methods you've been taught for finding their zeros. If there is no factor given, you can start by making use of the rational root theorem. Here, it tells you that rational roots will be factors of -12.

Descartes' rule of signs tells you there are 3 or 1 positive real roots and no negative real roots.

A function of odd degree will always have at least one real root, so finding that is a place to start.

answer: the common difference is -0.10