Two 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 find the sum of these two polynomials, I combined like terms: $$C(x) = A(x) + B(x) = 4x^4 + (-3x^3 + 2x^3) + 6x^2 + (2x) + (-7 + 1) The highest degree term in $C(x)$ is $4x^4$, which means the degree of this polynomial is 4.$$Which I then simplified to $$C(x) = 4x^4 - x^3 + 6x^2 + 2x – 6$$. Next, I worked on finding the difference between the two polynomials.

Question

For this, I did: $$D(x) = A(x) - B(x) = 4x^4 + (-3x^3 - 2x^3) + 6x^2 + (2x) + (-7 - 1) $$ Then, I moved on to the product of the two polynomials. Then I found that: $$E(x) = A(x) \ ⋅ B(x)$$ would give me a polynomial with a degree of 7 since the degree of $A(x)$ is 4 and the degree of $B(x)$ is 3. I then added those degrees together. Finally, I looked at the quotient of the two polynomials, $Q(x) = \{A(x)} {B(x)} $. Since the degree of $A(x)$ is higher than that of $B(x)$, I used polynomial long division. I found that the result would be of degree 1, as the leading term of $A(x)$ divided by the leading term of $B(x)$ gives me an $x$ term. I can derive the exact expressions for $E(x)$ and $Q(x)$ through detailed multiplication and division, but for now, I understand their degrees and how they relate to each other.

rewrite this in a clear form

GPT-4o mini · Answer

Here is a clearer version of your explanation regarding the operations performed on the polynomials $A(x)$ and $B(x)$:

Consider the polynomial expressions:

$A(x) = 4x^5 - 3x^3 + 2x - 7$ (degree 5)
$B(x) = 2x^3 + 6x^2 + 1$ (degree 3)

1. Sum of the Polynomials:

To find the sum $C(x) = A(x) + B(x)$, I combined like terms:

\[ C(x) = A(x) + B(x) = (4x^5) + (-3x^3 + 2x^3) + (6x^2) + (2x) + (-7 + 1) \]

Simplifying this, I have:

\[ C(x) = 4x^5 - x^3 + 6x^2 + 2x - 6 \]

The highest degree term in $C(x)$ is $4x^5$, so the degree of this polynomial is 5.

2. Difference of the Polynomials:

Next, I calculated the difference $D(x) = A(x) - B(x)$:

\[ D(x) = A(x) - B(x) = (4x^5) + (-3x^3 - 2x^3) + (6x^2) + (2x) + (-7 - 1) \]

This simplifies to:

\[ D(x) = 4x^5 - 5x^3 + 6x^2 + 2x - 8 \]

The highest degree term here is also $4x^5$, indicating that the degree of $D(x)$ is 5.

3. Product of the Polynomials:

For the product of the two polynomials $E(x) = A(x) \cdot B(x)$, I determined that the degree of the resulting polynomial is 8. Since the degree of $A(x)$ is 5 and the degree of $B(x)$ is 3, I add the degrees:

\[ \text{Degree of } E(x) = 5 + 3 = 8 \]

4. Quotient of the Polynomials:

Finally, I looked at the quotient $Q(x) = \frac{A(x)}{B(x)}$. Given that the degree of $A(x)$ (5) is higher than that of $B(x)$ (3), I applied polynomial long division. The leading term of the quotient will be $x^2$ since dividing the leading term of $A(x)$ (which is $4x^5$) by the leading term of $B(x)$ (which is $2x^3$) results in:

\[ \frac{4x^5}{2x^3} = 2x^2 \]

Thus, the degree of $Q(x)$ is 2.

In summary, the degrees of the resultant polynomials are as follows:

Degree of $C(x)$: 5
Degree of $D(x)$: 5
Degree of $E(x)$: 8
Degree of $Q(x)$: 2

This version organizes the information more clearly and separates the different operations for easier understanding.