Describe the key features of g(x), including the end behavior, y-intercept, and zeros of g(x) = x^3 - 3x^2 - 4x + 12. Then create a graph of the polynomial function.

Question

describe end behavior, y intercept, and zeros then create a graph for that function.

Answer 1

g(x) = x^3 - 3x^2-4x + 12

Note that the third and fourth coefficients are -4 * the first two.

We thus see that the polynomial can be factored as (x-3)(x^2-4) = (x-3)(x-2)(x+2)

As the leading coefficient is positive, and the power of x for this term is odd, g(x) goes to infinity as x goes to infinity and - infinity as x goes to - infinity.

The constant coefficient is 12, so the y intercept is 12.

From factoring the polynomial, we know that the 0s are -2, 2, and 3.

The polynomial is negative on (-∞, -2) and (2, 3) and positive on (-2, 2) and (3, ∞). Knowing the 0s, none is a multiple root, and where the function goes to infinity and negative infinity tells us this; you can also get this by plugging in x in each of these intervals.

A graph from -4 to 5 probably works well. In addition to calculating values at the integers, you probably want to calculate at -1/2 and 2 1/2.