Question

Revise this work : Polynomial #1 (P(x) = 2x^3 + 3x^2 - x + 4) Polynomial #2 = Q(x) = 5x^4 - 2x + 7)

Sum : R(x) = (2x^3 + 3x^2 - x + 4) + (5x^4 - 2x +7)
R(x) = 5x^4 + 2x^3 + 3x^2 - 3x + 11

Difference : S(x) = 5x^4 - 2x + 7) - (2x^3 + 3x^2 - x + 4) S(x) = 5x^4 - 2x - 7 - 2x^3 - 3x^2 + x - 4
S(x) = 5x^4 - 2x^3 - 3x^2 - x + 3

Product : T(x) = 10x^7 + 15x^6 - 5x^5 + 16x^4 + 8x^3 + 23x^2 - 15x + 28

Quotient : U(x) = (5x^4 - 2x + 7) ⁄ (2x^3 + 3x^2 - x + 4)

Answer 1

Here’s a revised version of your work, with improved clarity and formatting.

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### Polynomial Operations

**Polynomial #1**:
\( P(x) = 2x^3 + 3x^2 - x + 4 \)

**Polynomial #2**:
\( Q(x) = 5x^4 - 2x + 7 \)

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**Sum**:
To find the sum R(x):
\[
R(x) = P(x) + Q(x) = (2x^3 + 3x^2 - x + 4) + (5x^4 - 2x + 7)
\]
Combining like terms:
\[
R(x) = 5x^4 + 2x^3 + 3x^2 - 3x + 11
\]

---

**Difference**:
To find the difference S(x):
\[
S(x) = Q(x) - P(x) = (5x^4 - 2x + 7) - (2x^3 + 3x^2 - x + 4)
\]
Expanding the expression:
\[
S(x) = 5x^4 - 2x + 7 - 2x^3 - 3x^2 + x - 4
\]
Combining like terms:
\[
S(x) = 5x^4 - 2x^3 - 3x^2 - x + 3
\]

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**Product**:
To find the product T(x):
\[
T(x) = P(x) \cdot Q(x) = (2x^3 + 3x^2 - x + 4)(5x^4 - 2x + 7)
\]
Calculating the product:
\[
T(x) = 10x^7 + 15x^6 - 5x^5 + 16x^4 + 8x^3 + 23x^2 - 15x + 28
\]

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**Quotient**:
To find the quotient U(x):
\[
U(x) = \frac{Q(x)}{P(x)} = \frac{5x^4 - 2x + 7}{2x^3 + 3x^2 - x + 4}
\]
(Note: This expression would typically be simplified using polynomial long division or synthetic division, depending on the requirement.)

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This revised presentation organizes the information clearly and maintains the mathematical integrity of the original expressions.