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Bot GPT 4-o mini Let's choose two polynomial expressions as follows:
1. Polynomial \( P(x) = 2x^3 + 3x^2 - x + 4 \) (degree 3)
2. Polynomial \( Q(x) = 5x^4 - 2x + 7 \) (degree 4)
### 1. Sum of the Polynomials
To find the sum \( R(x) = P(x) + Q(x) \):
\[
R(x) = (2x^3 + 3x^2 - x + 4) + (5x^4 - 2x + 7)
\]
Combining like terms:
\[
R(x) = 5x^4 + 2x^3 + 3x^2 - 3x + 11
\]
### 2. Difference of the Polynomials
To find the difference \( S(x) = Q(x) - P(x) \):
\[
S(x) = (5x^4 - 2x + 7) - (2x^3 + 3x^2 - x + 4)
\]
Combining like terms:
\[
S(x) = 5x^4 - 2x - 7 - 2x^3 - 3x^2 + x - 4
\]
\[
S(x) = 5x^4 - 2x^3 - 3x^2 - x + 3
\]
### 3. Product of the Polynomials
To find the product \( T(x) = P(x) \times Q(x) \):
Multiply each term in \( P(x) \) by each term in \( Q(x) \):
\[
T(x) = (2x^3 + 3x^2 - x + 4)(5x^4 - 2x + 7)
\]
This will yield:
1. \( 2x^3 \times 5x^4 = 10x^7 \)
2. \( 2x^3 \times -2x = -4x^4 \)
3. \( 2x^3 \times 7 = 14x^3 \)
4. \( 3x^2 \times 5x^4 = 15x^6 \)
5. \( 3x^2 \times -2x = -6x^3 \)
6. \( 3x^2 \times 7 = 21x^2 \)
7. \( -x \times 5x^4 = -5x^5 \)
8. \( -x \times -2x = 2x^2 \)
9. \( -x \times 7 = -7x \)
10. \( 4 \times 5x^4 = 20x^4 \)
11. \( 4 \times -2x = -8x \)
12. \( 4 \times 7 = 28 \)
Now, combining these terms:
\[
T(x) = 10x^7 + 15x^6 - 5x^5 + (20x^4 - 4x^4) + (14x^3 - 6x^3) + (21x^2 + 2x^2) + (-7x - 8x) + 28
\]
\[
T(x) = 10x^7 + 15x^6 - 5x^5 + 16x^4 + 8x^3 + 23x^2 - 15x + 28
\]
### 4. Quotient of the Polynomials
To find the quotient \( U(x) = \frac{Q(x)}{P(x)} \):
This division of polynomials does not yield a simple polynomial, so we'll represent it in terms of the division process without actually performing polynomial long division. The result will be of the form:
\[
U(x) = \frac{5x^4 - 2x + 7}{2x^3 + 3x^2 - x + 4}
\]
In summary, we have the following:
- **Sum:** \( R(x) = 5x^4 + 2x^3 + 3x^2 - 3x + 11 \)
- **Difference:** \( S(x) = 5x^4 - 2x^3 - 3x^2 - x + 3 \)
- **Product:** \( T(x) = 10x^7 + 15x^6 - 5x^5 + 16x^4 + 8x^3 + 23x^2 - 15x + 28 \)
- **Quotient:** \( U(x) = \frac{5x^4 - 2x + 7}{2x^3 + 3x^2 - x + 4} \)