- The polynomial \(P(x)\) is expressed in factored form as \(P(x) = k \cdot x(x+3)(x-3)^2\) based on its given roots.
- Using the point \((5, 96)\), the equation \(P(5) = k \cdot 5(8)(2)^2 = 160k = 96\) is solved for \(k\), yielding \(k = 0.6\).
- The final formula for the polynomial is \(P(x) = 0.6x(x+3)(x-3)^2\).
- The final answer is \(\boxed{P(x) = 0.6x(x+3)(x-3)^2}\).
Explanation
- Analyzing the Roots and Form of P(x)
We are looking for a polynomial \(P(x)\) of degree 4. We are given its roots and their multiplicities:
- A root of multiplicity 2 at \(x = 3\), which implies that \((x - 3)^2\) is a factor.
- A root of multiplicity 1 at \(x = 0\), which implies that \(x\) is a factor.
- A root of multiplicity 1 at \(x = -3\), which implies that \((x + 3)\) is a factor.
Given these, we can express the polynomial as:
\[P(x) = k \cdot x \cdot (x + 3) \cdot (x - 3)^2\]
where \(k\) is a constant to be determined. The degree of this polynomial is \(1 + 1 + 2 = 4\), which matches the problem statement.
- Determining the Constant k
We are also given that the polynomial passes through the point \((5, 96)\), meaning \(P(5) = 96\). We can substitute \(x = 5\) into our general form to find the constant \(k\):
\[P(5) = k \cdot 5 \cdot (5 + 3) \cdot (5 - 3)^2 = 96\]
First, simplify the terms:
- \(5 + 3 = 8\)
- \(5 - 3 = 2\)
- \((5 - 3)^2 = 2^2 = 4\)
Substituting these back in:
\[k \cdot 5 \cdot 8 \cdot 4 = 96\]
\[k \cdot 160 = 96\]
Dividing both sides by 160:
\[k = \frac{96}{160} = \frac{6}{10} = 0.6\]
Thus, the constant \(k\) is \(0.6\).
- Finalizing the Polynomial Formula
Now that we have found \(k = 0.6\), we can write the final formula for \(P(x)\):
\[P(x) = 0.6 \cdot x \cdot (x + 3) \cdot (x - 3)^2\]
We can expand this if desired, though the factored form is already a valid formula. Let's provide the final expression:
\[P(x) = 0.6x(x+3)(x-3)^2\]
This is the polynomial of degree 4 satisfying all the given conditions.
\[\boxed{P(x) = 0.6x(x+3)(x-3)^2}\]
Examples
Polynomials are widely used in modeling real-world phenomena. For example, if you were modeling the trajectory of a projectile or the growth of a population over time, you would often use a polynomial to represent the relationship between variables. In this case, finding the formula of \(P(x)\) given its roots and a specific point allows us to reconstruct a complete mathematical model from partial data, a fundamental skill in data fitting and engineering.