Cheak whether `7+3x` is a factor of `3x^(3)+7x` .

Answer

To check whether \( 7 + 3x \) is a factor of \( 3x^3 + 7x \), we can use the Remainder Theorem. According to this theorem, if we divide a polynomial \( P(x) \) by \( (x - c) \), the remainder of this division is \( P(c) \). If the remainder is zero, then \( (x - c) \) is a factor of \( P(x) \).

### Step-by-Step Solution:

1. **Identify the Polynomial and the Divisor**:
- The polynomial \( P(x) = 3x^3 + 7x \).
- The divisor is \( 7 + 3x \). We can rewrite this as \( 3x + 7 \).

2. **Set the Divisor to Zero**:
- To find the value of \( x \) that makes the divisor zero, we set:
\[ 3x + 7 = 0 \]
- Solving for \( x \):
\[ 3x = -7 \implies x = -\frac{7}{3} \]

3. **Substitute \( x = -\frac{7}{3} \) into the Polynomial**:
- We need to evaluate \( P\left(-\frac{7}{3}\right) \):
\[ P\left(-\frac{7}{3}\right) = 3\left(-\frac{7}{3}\right)^3 + 7\left(-\frac{7}{3}\right) \]

4. **Calculate \( \left(-\frac{7}{3}\right)^3 \)**:
- First, calculate \( \left(-\frac{7}{3}\right)^3 \):
\[ \left(-\frac{7}{3}\right)^3 = -\frac{343}{27} \]
- Now substitute this back into the polynomial:
\[ P\left(-\frac{7}{3}\right) = 3\left(-\frac{343}{27}\right) + 7\left(-\frac{7}{3}\right) \]

5. **Simplify the Expression**:
- Calculate \( 3\left(-\frac{343}{27}\right) \):
\[ = -\frac{1029}{27} \]
- Now calculate \( 7\left(-\frac{7}{3}\right) \):
\[ = -\frac{49}{3} = -\frac{441}{27} \quad (\text{converting to a common denominator}) \]
- Now combine the two results:
\[ P\left(-\frac{7}{3}\right) = -\frac{1029}{27} - \frac{441}{27} = -\frac{1470}{27} \]

6. **Determine the Remainder**:
- The remainder is \( -\frac{1470}{27} \). Since this is not zero, we conclude that \( 7 + 3x \) is not a factor of \( 3x^3 + 7x \).

### Conclusion:
Since the remainder is not zero, we can conclude that \( 7 + 3x \) is **not** a factor of \( 3x^3 + 7x \).