Questions: Find a possible formula for the polynomial shown on the right. NOTE: Enter the formula with lowest possible degree. y=

Find a possible formula for the polynomial shown on the right. NOTE: Enter the formula with lowest possible degree. y=
Transcript text: Find a possible formula for the polynomial shown on the right. NOTE: Enter the formula with lowest possible degree. \[ y=\square \]
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Solution

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Solution Steps

Step 1: Identifying the roots and their multiplicity

The graph crosses the x-axis at x = 0, x = 4, and x = -1. The graph touches the x-axis at x = 2. Therefore, the roots are x = 0, x = 4, x = -1, and x = 2 (with a multiplicity of 2).

Step 2: Constructing the polynomial

A possible formula for the polynomial is obtained by multiplying the factors corresponding to each root. Since we want the lowest possible degree, we use the smallest multiplicity for each root:

f(x) = x(x-4)(x+1)(x-2)^2

The leading coefficient can be adjusted. Since the graph passes through the point (1,6), we substitute these values in:

6 = 1(1-4)(1+1)(1-2)^2
6 = (1)(-3)(2)(-1)^2
6 = -6

We need to multiply the whole function by -1.

Final Answer:

f(x) = -x(x-4)(x+1)(x-2)²

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