Question

The degree of the polynomial function f(x) is 3. The roots of the equation f(x)=0 are −4 , 0, and 2. Which graph could be the graph of f(x) ?

Answer 1

We know that the polynomial function is of degree 3, and that its roots are -4, 0, 2.

With this data we can write a generic equation for the function:

f (x) = bx (x + 4) (x-2)

Since the function is of degree 3 and cuts the axis at x = 0, then it has rotational symmetry with respect to the origin.

The graph of the function can be of two main forms, based on the value of the coefficient b.

If b is positive then the function grows from y = -infinite and cuts the x-axis for the first time in -4. Then it decreases, cuts at x = 0 and begins to grow again cutting the x-axis for the third time at x = 2. and continues to grow until y = infnit

If b is negative, then the function decreases from y = infinity and cuts the x-axis for the first time in -4. Then it grows, cuts at x = 0 and begins to decrease again by cutting the x-axis for the third time at x = 2, and continues to decrease until y = -infnit.

In the attached images the graphs of the function f (x) are shown assuming b = -1 and b = 1

Answer 2

Given the zeros of a polynomial as -4, 0 and 2. The factors of the polynomial function will be (x+4)(x-0)(x-2).

The polynomial function will be the product of the factors.

Also, the zeros show that the polynomial curve will cut the x axis at the given points.

Get the polynomial function first

P(x) =  (x+4)(x-0)(x-2).

P(x) = (x²+4x)(x-2)

P(x) = x³-2x²+4x²-8x

P(x) = x³+2x²-8x

Plotting the graph of the function as shown below:

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