Question
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.
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
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
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GPT-4o mini
Here is a clearer version of your explanation regarding the operations performed on the polynomials \(A(x)\) and \(B(x)\):
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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
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This version organizes the information more clearly and separates the different operations for easier understanding.
---
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.
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