Use the factors to identify the zeros of f(x)=3x^3+12x^2−36x. Then sketch the graph of the polynomial.

York

York

Answered question

2021-06-21

Use the factors to identify the zeros of f(x)=3x3+12x236x. Then sketch the graph of the polynomial.

Answer & Explanation

Isma Jimenez

Isma Jimenez

Skilled2021-06-22Added 84 answers

The terms in f(x)=3x3+12x236x have a common factor of 3x. Factoring out 3x then gives f(x)=3x(x2+4x12)). Factoring the quadratic x2+4x12x gives (x+6)(x2)(x+6)(x2) so f(x)=3x(x+6)(x2)f(x)=3x(x+6)(x2). Setting each factor equal to 0 and solving for x then gives zeros of:
3x=0x+6=0x2=0
x=0x=6x=2
Since f(x) has an odd degree of 3, the ends will go in opposite directions. Since f(x) has a positive leading coefficient of 3, the graph will rise to the right and fall to the left. Since the zeros all have an odd multiplicity of 1 (since none of the factors repeated), then the graph will cross the xx-axis at the zeros. Plot the zeros of −6, 0, and 2. Then sketch a function that passes through the xx-axis at the zeros, rises to the right, and falls to the left:
[Graph]

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