Create a quintic polynomial inequality for which x=-4, x=0, x ≥ 2 is the solution. Justify your answer by solving the inequality using chart. Provide a diagram as well
Your wording is identical and as murky as a previous post, so I'll give you the previous answer.
https://www.jiskha.com/questions/1829039/create-a-quintic-polynomial-inequality-for-which-x-4-x-0-x-2-is-the-solution#2029800
To create a quintic polynomial inequality for which x = -4, x = 0, and x ≥ 2 is the solution, we can start by considering the given solutions and their properties.
First, we have x = -4, which means the polynomial should have a factor of (x + 4). Similarly, x = 0 implies a factor of x, and x ≥ 2 indicates that the polynomial should be greater than or equal to zero for x greater than or equal to 2.
Let's start by constructing the polynomial using these factors:
p(x) = a(x + 4)(x)(x - 2)(x - r)(x - s)
Here, a is a constant, and r and s are the remaining two unknown roots.
The polynomial has a degree of five (quintic) since it has five factors.
To determine the remaining two roots, r and s, let's consider their properties based on the given information. Since x = -4 is a solution, we substitute it into the polynomial:
p(-4) = a(-4 + 4)(-4)(-4 - 2)(-4 - r)(-4 - s)
Since (-4 + 4) and (-4)(-4 - 2) are both equal to zero, the term a(-4 + 4)(-4)(-4 - 2) simplifies to zero. Hence:
p(-4) = 0
This means that the factor (-4 - r)(-4 - s) should also be equal to zero, as any factor multiplying zero results in zero. Therefore, one of the two remaining factors must be -4.
Without loss of generality, we can assume r = -4.
So now our polynomial is:
p(x) = a(x + 4)(x)(x - 2)(x + 4)(x - s)
To determine the value of s, we use the fact that x = 0 is a solution:
p(0) = a(0 + 4)(0)(0 - 2)(0 + 4)(0 - s)
Once again, since (0 + 4)(0)(0 - 2) is equal to zero, the entire product becomes zero. Hence, we have:
p(0) = 0
From this, we can deduce that s must be zero, without loss of generality.
Now our polynomial becomes:
p(x) = a(x + 4)(x)(x - 2)(x + 4)(x)
To ensure the inequality holds true, we can adjust the value of a to control the behavior of the polynomial.
Since we want the solution to be x ≥ 2, the graph of the polynomial should be greater than or equal to zero for x ≥ 2.
For simplicity, let's assume a = 1. Then, the polynomial becomes:
p(x) = (x + 4)(x)(x - 2)(x + 4)(x)
We can now construct a chart to determine the behavior of the polynomial:
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| x | -∞ | -4 | 0 | 2 | +∞ |
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| p(x)| (-) | 0 | 0 | + | + |
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Based on the chart, the polynomial p(x) is negative to the left of -4, zero at x = -4 and x = 0, and positive for x ≥ 2. This aligns with the desired solution.
Now let's provide a diagram to illustrate the behavior of the polynomial:
+
|
|
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-----0---o---o---+-------------------------
-4 0 2
The plus signs indicate positivity, and the line indicates non-negativity (zero).
As shown in the diagram, the polynomial is negative to the left of -4, zero at x = -4 and x = 0, and positive for x ≥ 2.
Therefore, the quintic polynomial inequality that satisfies the given solution is:
p(x) ≥ 0 where p(x) = (x + 4)(x)(x - 2)(x + 4)(x)