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.

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? 
short

GPT-4o mini · Answer

### Polynomial Expressions: 1. **Polynomial (Degree 3): $$ P(x) = 2x^3 - 4x^2 + 5 $$ 2. **Polynomial B** (Degree 4): $$ Q(x) = x^4 + 3x^3 - 2x + 6 $$ ### Operations on the Polynomials: 1. **Sum** $ P(x) + Q(x) $: $$ R(x) = P(x) + Q(x) = (2x^3 - 4x^2 + 5) + (x^4 + 3x^3 - 2x + 6) = x^4 + (2x^3 + 3x^3) - 4x^2 - 2x + (5 + 6) = x^4 + 5x^3 - 4x^2 - 2x + 11 $$ 2. **Difference** $ P(x) - Q(x) $: $$ S(x) = P(x) - Q(x) = (2x^3 - 4x^2 + 5) - (x^4 + 3x^3 - 2x + 6) = -x^4 + (2x^3 - 3x^3) - 4x^2 + 2x + (5 - 6) = -x^4 - x^3 - 4x^2 + 2x - 1 $$ 3. **Product** $ P(x) \cdot Q(x) $: $$ T(x) = P(x) \cdot Q(x) = (2x^3 - 4x^2 + 5)(x^4 + 3x^3 - 2x + 6) $$ (Expanding this polynomial would require further calculation, but we know the final degree will be 7, as 3 + 4 = 7.) 4. **Quotient** $ \frac{Q(x)}{P(x)} $: $$ U(x) = \frac{Q(x)}{P(x)} = \frac{x^4 + 3x^3 - 2x + 6}{2x^3 - 4x^2 + 5} $$ (Performing polynomial long division here would show that the leading term of the quotient is $\frac{x^4}{2x^3} = \frac{1}{2} x$.) --- ### Response to Classmates: #### Classmate 1: - **Finding the Sum:** Assume their polynomials are $ A(x) $ and $ B(x) $. The sum is calculated straightforwardly by adding the coefficients of the same degree terms. - **Rules for First and Last Terms:** - First term: Identified as the leading coefficient and variable term with the highest degree from either polynomial. - Last term: The constant term found by adding the constant terms of both polynomials. - **Limitations:** This approach only works if you’re only adding. If there are negative components, the signs must be incorporated, which is not accounted for here. #### Classmate 2: - **Finding the Product:** The product can be found by distributing each term of the first polynomial across all terms of the second polynomial. - **Rules for First and Last Terms:** - First term: The leading term comes from multiplying the leading term of the first polynomial with the leading term of the second. - Last term: The constant term is obtained by multiplying the constant terms of both polynomials. - **Limitations:** This method assumes both polynomials are positive leading terms. Negative terms would affect the sign of the resulting product, which should be carefully treated.