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

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?
a short answer, please

Answer 1

Polynomial Expressions

Let:

• Polynomial \( P(x) = 3x^4 + 2x^3 - 5x + 7 \) (Degree 4)
• Polynomial \( Q(x) = 2x^3 - 4x^2 + x - 1 \) (Degree 3)

Operations on the Polynomials

1. Sum:
\[ P(x) + Q(x) = (3x^4 + 2x^3 - 5x + 7) + (2x^3 - 4x^2 + x - 1) \]
\[ = 3x^4 + (2x^3 + 2x^3) - 4x^2 + (-5x + x) + (7 - 1) \]
\[ = 3x^4 + 4x^3 - 4x^2 - 4x + 6 \]

2. Difference:
\[ P(x) - Q(x) = (3x^4 + 2x^3 - 5x + 7) - (2x^3 - 4x^2 + x - 1) \]
\[ = 3x^4 + (2x^3 - 2x^3) + 4x^2 + (-5x - x) + (7 + 1) \]
\[ = 3x^4 + 4x^2 - 6x + 8 \]

3. Product:
\[ P(x) \cdot Q(x) = (3x^4 + 2x^3 - 5x + 7)(2x^3 - 4x^2 + x - 1) \]
This will yield terms of varying degrees. However, the leading term will be:
\[ 3x^4 \cdot 2x^3 = 6x^7 \]
Thus, the product is:
\[ 6x^7 + \text{(other terms)} \]

4. Quotient:
\[ \frac{P(x)}{Q(x)} = \frac{3x^4 + 2x^3 - 5x + 7}{2x^3 - 4x^2 + x - 1} \]
The leading term of the quotient yields:
\[ \frac{3x^4}{2x^3} = \frac{3}{2}x \]
For the final quotient, more division would be needed, though the leading term stands.

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Responses to Classmates

Response to Classmate 1:

Classmate x(x) = 5x^3 + 3x^2 - 2

Sum:
To find the sum of your polynomial and mine, add the coefficients of matching degrees. The first term will come from the highest degree (here 5x^3) and the last term will be the constant term from both polynomials. The resulting polynomial will have a degree equal to the higher of the two.

Rules:

1. For the first term, take the term with the highest degree in either polynomial.
2. For the last term, sum the constant terms of both polynomials.

Limitations: This method assumes both polynomials are properly aligned and that there are no like terms making unexpected combinations.

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Response to Classmate 2:

Classmate y(x) = 4x^2 + 1 - x^3

Difference:
To calculate the difference between your polynomial and mine, subtract the coefficient of your highest degree term from mine. The first term will be derived from the higher degree polynomial and the last term will again be calculated from the constant terms.

Rules:

1. For the first term, again look for the polynomial with the highest degree and record its leading term.
2. For the last term, subtract the constant terms similarly.

Limitations: This method can lead to an oversight of sign changes if not careful while subtracting, particularly when polynomials are complex.