If sum of two polynomials is `5x^(2)+3x-1`. If one of them is `3x^(3)-2x+7`. Find the other.

Options

A. `3x^(3)-5x^(2)+5x-8`
B. `-3x^(3)+5x^(2)+5x-8`
C. `2x^(2)+x+6`
D. none of these

Answer

To find the other polynomial, we can use the information given in the problem. We know that the sum of two polynomials is equal to a specific polynomial, and we have one of the polynomials. We can find the other polynomial by rearranging the equation.

### Step-by-Step Solution:

1. **Write down the equation for the sum of the polynomials:**
Let \( P(x) \) be the first polynomial and \( Q(x) \) be the second polynomial. According to the problem:
\[ P(x) + Q(x) = 5x^2 + 3x - 1 \]
We know that \( P(x) = 3x^3 - 2x + 7 \).

2. **Substitute \( P(x) \) into the equation:**
Replace \( P(x) \) in the equation with the polynomial we have:
\[ (3x^3 - 2x + 7) + Q(x) = 5x^2 + 3x - 1 \]

3. **Isolate \( Q(x) \):**
To find \( Q(x) \), we need to isolate it on one side of the equation. We can do this by subtracting \( P(x) \) from both sides:
\[ Q(x) = (5x^2 + 3x - 1) - (3x^3 - 2x + 7) \]

4. **Simplify the right side:**
Now, we will simplify the expression on the right side:
\[ Q(x) = 5x^2 + 3x - 1 - 3x^3 + 2x - 7 \]
Combine like terms:
\[ Q(x) = -3x^3 + 5x^2 + (3x + 2x) + (-1 - 7) \]
\[ Q(x) = -3x^3 + 5x^2 + 5x - 8 \]

5. **Final result:**
Therefore, the other polynomial \( Q(x) \) is:
\[ Q(x) = -3x^3 + 5x^2 + 5x - 8 \]