Question

Find the degree of the polynomial that produces these successive results: 24, 0, -24, 0, 144, 504

Answer 1

To find the degree of the polynomial that produces the given values \(24, 0, -24, 0, 144, 504\), we can begin by analyzing the structure of the outputs. We will denote the function that produces these values as \(f(x)\), with specific values given at specific points.

To facilitate the analysis, let's label the outputs:

- Let \(f(0) = 24\)
- Let \(f(1) = 0\)
- Let \(f(2) = -24\)
- Let \(f(3) = 0\)
- Let \(f(4) = 144\)
- Let \(f(5) = 504\)

### Step 1: Identifying Roots
From the above values, we observe that \(f(1) = 0\) and \(f(3) = 0\). This implies that \(1\) and \(3\) are roots of the polynomial \(f(x)\).

### Step 2: Formulating Initial Polynomial
Since \(f(1) = 0\) and \(f(3) = 0\), we can express \(f(x)\) as:
\[
f(x) = (x-1)(x-3)g(x),
\]
where \(g(x)\) is another polynomial that we need to determine.

### Step 3: Evaluating at Known Points
We can also evaluate \(f(x)\) at the other points. Note the given outputs and establish a system to solve for the coefficients in \(g(x)\).

### Higher Degree Polynomial
Given that we have six points and know two roots already, it is possible that \(g(x)\) is a polynomial of degree \(n-2\), where \(n\) is the degree of the original polynomial. This suggests that the total degree could be larger. Let's assume \(g(x)\) is also a polynomial of degree \(4-k\), which could at least be \(1\) because we want to fit all values.

### Step 4: Guessing Polynomial
Following our test with values up to \(f(4) = 144\) and \(f(5) = 504\), it suggests higher degrees or non-linear contributions should be observed.

To directly evaluate existing polynomial values:
- To fit a polynomial to 6 points which yields the given outputs, we will typically express this as a polynomial of degree \(5\).

### Conclusion
Since there are \(6\) unique outputs from \(f(0)\) to \(f(5)\), by polynomial interpolation, particularly considering the oscillating pattern noted (including zeros and changes in sign), we can ascertain that the minimum degree of \(f(x)\) is \(5\).

\[
\text{The degree of the polynomial } f(x) \text{ is, therefore, } 5.
\]