Question
find the degree of the polynomial that produces these successive results: 24, 0 , -24 ,0, 144, 504
Answers
GPT-4o mini
To determine the degree of the polynomial that produces the given sequence of outputs (24, 0, -24, 0, 144, 504), we can observe the pattern in the results.
The outputs seem to vary in a way that suggests a polynomial function might be involved. Since we have six outputs, we can expect the polynomial could be a degree 5 polynomial, as a polynomial of degree \( n \) can have at most \( n \) roots. However, we must verify this.
First, let's denote the outputs as \( P(0) = 24 \), \( P(1) = 0 \), \( P(2) = -24 \), \( P(3) = 0 \), \( P(4) = 144 \), and \( P(5) = 504 \).
### Step 1: Identify the Roots
The polynomial is zero at \( x = 1 \) and \( x = 3 \):
- \( P(1) = 0 \)
- \( P(3) = 0 \)
### Step 2: Listing Roots and Values
Thus, we know that the polynomial can be expressed as:
\[
P(x) = k(x - 1)(x - 3)Q(x)
\]
where \( Q(x) \) is another polynomial.
### Step 3: Evaluate Additional Values
We can look at other values to figure out information about \( Q(x) \):
1. \( P(0) = 24 \)
\[
P(0) = k(-1)(-3)Q(0) = 3k Q(0) = 24 \implies k Q(0) = 8
\]
2. \( P(2) = -24 \)
\[
P(2) = k(2 - 1)(2 - 3)Q(2) = k(1)(-1)Q(2) = -kQ(2) = -24 \implies k Q(2) = 24
\]
3. \( P(4) = 144 \)
\[
P(4) = k(4 - 1)(4 - 3)Q(4) = k(3)(1)Q(4) = 3k Q(4) = 144 \implies k Q(4) = 48
\]
4. \( P(5) = 504 \)
\[
P(5) = k(5 - 1)(5 - 3)Q(5) = k(4)(2)Q(5) = 8k Q(5) = 504 \implies k Q(5) = 63
\]
### Step 4: Finding Degree
We see that \( P(x) \) has roots and the coefficients can be solved. Moreover, the highest output values also indicate its degree. Given the polynomial has its structure showing at least two roots confirmed and being computed for values, this suggests that it could be at least degree 4.
### Conclusion
In summary, given that the polynomial has roots at \( x = 1 \) and \( x = 3 \) with varied results at spaced intervals and having a functional continuing at entrances, we can confidently determine that the polynomial degree must be **at least 5** for uniqueness to hold over the distinct values provided. Thus, the **degree of the polynomial is 5** suggesting \( P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \).
The outputs seem to vary in a way that suggests a polynomial function might be involved. Since we have six outputs, we can expect the polynomial could be a degree 5 polynomial, as a polynomial of degree \( n \) can have at most \( n \) roots. However, we must verify this.
First, let's denote the outputs as \( P(0) = 24 \), \( P(1) = 0 \), \( P(2) = -24 \), \( P(3) = 0 \), \( P(4) = 144 \), and \( P(5) = 504 \).
### Step 1: Identify the Roots
The polynomial is zero at \( x = 1 \) and \( x = 3 \):
- \( P(1) = 0 \)
- \( P(3) = 0 \)
### Step 2: Listing Roots and Values
Thus, we know that the polynomial can be expressed as:
\[
P(x) = k(x - 1)(x - 3)Q(x)
\]
where \( Q(x) \) is another polynomial.
### Step 3: Evaluate Additional Values
We can look at other values to figure out information about \( Q(x) \):
1. \( P(0) = 24 \)
\[
P(0) = k(-1)(-3)Q(0) = 3k Q(0) = 24 \implies k Q(0) = 8
\]
2. \( P(2) = -24 \)
\[
P(2) = k(2 - 1)(2 - 3)Q(2) = k(1)(-1)Q(2) = -kQ(2) = -24 \implies k Q(2) = 24
\]
3. \( P(4) = 144 \)
\[
P(4) = k(4 - 1)(4 - 3)Q(4) = k(3)(1)Q(4) = 3k Q(4) = 144 \implies k Q(4) = 48
\]
4. \( P(5) = 504 \)
\[
P(5) = k(5 - 1)(5 - 3)Q(5) = k(4)(2)Q(5) = 8k Q(5) = 504 \implies k Q(5) = 63
\]
### Step 4: Finding Degree
We see that \( P(x) \) has roots and the coefficients can be solved. Moreover, the highest output values also indicate its degree. Given the polynomial has its structure showing at least two roots confirmed and being computed for values, this suggests that it could be at least degree 4.
### Conclusion
In summary, given that the polynomial has roots at \( x = 1 \) and \( x = 3 \) with varied results at spaced intervals and having a functional continuing at entrances, we can confidently determine that the polynomial degree must be **at least 5** for uniqueness to hold over the distinct values provided. Thus, the **degree of the polynomial is 5** suggesting \( P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \).