Question

Write two examples of a polynomial expression:
Each with a minimum of three terms
A degree no larger than four.
Give each polynomial expression a different degree.
Then find the sum, difference, product, and quotient of your two polynomials. (4 different problems)
For the quotient, use the polynomial expression with the larger degree as the numerator.

Answer 1

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:** \( U(x) = \frac{5x^4 - 2x + 7}{2x^3 + 3x^2 - x + 4} \)