Question

Write two examples of a polynomial expression, each with a minimum of three terms and 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. For the quotient, use the polynomial expression with the larger degree as the numerator.

After you have posted your polynomial expressions and their sum, difference, product, and quotient, respond to two of your classmates' posts. In your response to each post, complete the following:

Find either the sum, difference, product, or quotient of your classmates' polynomial expressions. Choose only one operation; you do not need to perform all four.

Create rules for how to approach finding the first and last terms of your peers' polynomial expressions, without working them out fully. Explain how you derived your rules. Do any of these rules have limitations?

Answer 1

### Polynomial Expressions

**Polynomial 1 (Degree 3):**
\( P(x) = 2x^3 - 4x^2 + 6 \)

**Polynomial 2 (Degree 4):**
\( Q(x) = 3x^4 + 2x^3 - x + 5 \)

### Operations on the Polynomials

1. **Sum:**
\( P(x) + Q(x) = (2x^3 - 4x^2 + 6) + (3x^4 + 2x^3 - x + 5) \)
\[ = 3x^4 + (2x^3 + 2x^3) - 4x^2 - x + (6 + 5) \]
\[ = 3x^4 + 4x^3 - 4x^2 - x + 11 \]

2. **Difference:**
\( P(x) - Q(x) = (2x^3 - 4x^2 + 6) - (3x^4 + 2x^3 - x + 5) \)
\[ = -3x^4 + (2x^3 - 2x^3) - 4x^2 + x + (6 - 5) \]
\[ = -3x^4 - 4x^2 + x + 1 \]

3. **Product:**
\( P(x) \cdot Q(x) = (2x^3 - 4x^2 + 6)(3x^4 + 2x^3 - x + 5) \)
To simplify, we will state the highest degree term:
The leading term for \( P(x) \) is \( 2x^3 \) and for \( Q(x) \) is \( 3x^4 \).
The product will yield a leading term of:
\[ 2x^3 \cdot 3x^4 = 6x^{7} \]
(The remaining terms will produce degrees of \( 6 \), \( 5 \), \( 4 \), etc.)

4. **Quotient:**
\( \frac{Q(x)}{P(x)} = \frac{3x^4 + 2x^3 - x + 5}{2x^3 - 4x^2 + 6} \)
To find the quotient's leading term, we look for the ratio of the leading term of the numerator to the leading term of the denominator:
\[ \frac{3x^4}{2x^3} = \frac{3}{2}x \]
This indicates the first term of the quotient is \( \frac{3}{2}x \). The complete division will involve polynomial long division.

### Responses to Classmates' Posts

#### Post Example 1
**Classmate's Polynomial Expressions:**
- Polynomial A: \( 5x^3 - 3x + 2 \) (Degree 3)
- Polynomial B: \( 4x^2 + 6x + 1 \) (Degree 2)

**Operation:** Find the product of their polynomials.
**Product:**
\( (5x^3 - 3x + 2)(4x^2 + 6x + 1) \)
The leading term will be \( 5x^3 \cdot 4x^2 = 20x^5 \).

**Rules for First and Last Terms:**
To find the first term of the product of two polynomials, multiply the leading terms from each polynomial. The last term of the product can be found by multiplying the constant terms from each polynomial.
**Limitations:** This method works effectively unless polynomials have zero coefficients for higher degrees, which may affect the overall degree of the product.

---

#### Post Example 2
**Classmate's Polynomial Expressions:**
- Polynomial C: \( x^4 - 3x^2 + 7 \) (Degree 4)
- Polynomial D: \( 2x^3 + 5x - 1 \) (Degree 3)

**Operation:** Find the difference of their polynomials.
**Difference:**
\( (x^4 - 3x^2 + 7) - (2x^3 + 5x - 1) \)
This results in \( x^4 - 2x^3 - 3x^2 - 5x + 8 \).

**Rules for First and Last Terms:**
For the difference, the first term is the leading term from the polynomial with the higher degree, and the last term can be found by subtracting constant terms directly.
**Limitations:** This can be complicated with polynomials having leading coefficients that cancel out, changing the overall structure significantly.