A few examples of polynomial expressions are $A(x) = 4x^5 - 3x^3 + 2x - 7$ (which is of degree 5) and $B(x) = 2x^3 + 6x^2 + 1$ (which is of degree 3). To calculate the sum of these two polynomials, I added their like terms: $$C(x) = A(x) + B(x) = 4x^4 + (-3x^3 + 2x^3) + 6x^2 + (2x) + (-7 + 1). Next, I discovered the greatest degree term in $C(x)$ is $4x^4$, indicating the degree of this polynomial is 4. I then simplified it to \[C(x) = 4x^4 - x^3 + 6x^2 + 2x - 6$$. Next, I focused on determining the difference between the two polynomials. For this, I did: [D(x) = A(x) - B(x) = 4x^4 + (-3x^3 - 2x^3) + 6x^2 + (2x) +(-7 - 1) \] After that, I went on to the product of the two polynomials. Then I noticed that: $$E(x) = A(x) \ ⋅ B(x)$$ would give me a polynomial with a degree of 7 because the degree of $A(x)$ is 4 and the degree of $B(x)$ is 3. I combined the degrees. Finally, I examined the quotient between the two polynomials, $Q(x) = \{A(x)} {B(x)} $. Since $A(x)$ has a larger degree than $B(x)$, I utilized polynomial long division.

Question

A few examples of polynomial expressions are $A(x) = 4x^5 - 3x^3 + 2x - 7$ (which is of degree 5) and $B(x) = 2x^3 + 6x^2 + 1$ (which is of degree 3). To calculate the sum of these two polynomials, I added their like terms: $$C(x) = A(x) + B(x) = 4x^4 + (-3x^3 + 2x^3) + 6x^2 + (2x) + (-7 + 1). Next, I discovered the greatest degree term in $C(x)$ is $4x^4$, indicating the degree of this polynomial is 4. I then simplified it to \[C(x) = 4x^4 - x^3 + 6x^2 + 2x - 6$$. Next, I focused on determining the difference between the two polynomials.  For this, I did: [D(x) = A(x) - B(x) = 4x^4 + (-3x^3 - 2x^3) + 6x^2 + (2x) +(-7 - 1) \] After that, I went on to the product of the two polynomials. Then I noticed that: $$E(x) = A(x) \ ⋅ B(x)$$ would give me a polynomial with a degree of 7 because the degree of $A(x)$ is 4 and the degree of $B(x)$ is 3. I combined the degrees. Finally, I examined the quotient between the two polynomials, $Q(x) = \{A(x)} {B(x)} $. Since $A(x)$ has a larger degree than $B(x)$, I utilized polynomial long division.

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?

Explain how you agree with this post in your own words.

GPT-4o mini · Answer

Let's take the sum of two polynomial expressions as an example. Suppose we have two polynomial expressions from classmates:

$ F(x) = 5x^4 + 3x^2 - 4 $ (degree 4)
$ G(x) = -2x^3 + 7x + 5 $ (degree 3)

To find the sum $ H(x) = F(x) + G(x) $, we will add their like terms:

Step-by-step Approach:

Identify Like Terms: Group the terms of the same degree from both polynomials.
- $ F(x) $: $ 5x^4 + 0x^3 + 3x^2 + 0x - 4 $
- $ G(x) $: $ 0x^4 - 2x^3 + 0x^2 + 7x + 5 $
Add the Coefficients: Add the coefficients of each like term.
- For $ x^4 $: $ 5x^4 + 0 = 5x^4 $
- For $ x^3 $: $ 0 + (-2) = -2x^3 $
- For $ x^2 $: $ 3 + 0 = 3x^2 $
- For $ x $: $ 0 + 7 = 7x $
- Constant terms: $ -4 + 5 = 1 $
Combine the Results: Write the resulting polynomial in standard form by ordering the terms from highest degree to lowest.
- $ H(x) = 5x^4 - 2x^3 + 3x^2 + 7x + 1 $

Rules for Finding the First and Last Terms:

First Term:
- The first term of the resulting sum or polynomial (in standard form) will always be the term with the highest degree, which can be identified by checking the first polynomial and the second polynomial to find the maximum degree term.
- If both polynomials are of the same degree, simply add their leading coefficients to maintain the same degree.
Last Term:
- The last term (the constant term) in the resulting polynomial is found by adding the constant terms of both polynomials together.

Limitations:

If the polynomials have different degrees, the highest degree term always comes from the polynomial with the larger degree. However, if both polynomials contain terms of all degrees down to the constant, then the sum will also contain them all.
If there are no like terms in certain degrees, those terms simply carry through as $0$.

Agreement Explanation:

In summary, I agree with the overall approach and conclusion about how to work with polynomial expressions: by focusing on identifying and combining like terms correctly, you not only ensure that the final polynomial is accurate but also that you understand the operations performed on them, such as addition or subtraction. Recognizing how to first pick out the highest degree terms and how to finalize the expression will streamline the process significantly.