Let's choose two polynomial expressions as follows:
- Polynomial \( P(x) = 2x^3 + 3x^2 - x + 4 \) (degree 3)
- 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:
- \( 2x^3 \times 5x^4 = 10x^7 \)
- \( 2x^3 \times -2x = -4x^4 \)
- \( 2x^3 \times 7 = 14x^3 \)
- \( 3x^2 \times 5x^4 = 15x^6 \)
- \( 3x^2 \times -2x = -6x^3 \)
- \( 3x^2 \times 7 = 21x^2 \)
- \( -x \times 5x^4 = -5x^5 \)
- \( -x \times -2x = 2x^2 \)
- \( -x \times 7 = -7x \)
- \( 4 \times 5x^4 = 20x^4 \)
- \( 4 \times -2x = -8x \)
- \( 4 \times 7 = 28 \)
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: \( U(x) = \frac{5x^4 - 2x + 7}{2x^3 + 3x^2 - x + 4} \)