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.