To find the quotient of the polynomial 3x³ + 5x + 14 when divided by x², we can use polynomial long division.
1. **Set up the division**: We place the polynomial 3x³ + 5x + 14 under the long division symbol and x² outside it.
2. **Divide the leading terms**: Divide the leading term of the polynomial (3x³) by the leading term of the divisor (x²). This gives us:
- 3x³ ÷ x² = 3x
3. **Multiply**: Now, multiply the entire divisor (x²) by the result (3x):
- 3x * x² = 3x³
4. **Subtract**: Next, subtract this result from the original polynomial:
- (3x³ + 5x + 14) – (3x³) = 5x + 14
5. **Repeat the process**: Now, we will divide 5x by x². However, 5x divided by x² gives a result of 5/x (which cannot be simplified further), so we stop here.
Thus, the result of the long division can be expressed as:
- Q = 3x + (5x + 14)/x²
In conclusion, the quotient when dividing the polynomial 3x³ + 5x + 14 by x² is:
- **Quotient**: 3x with a remainder of **(5x + 14)/x²**
This means that:
- Completely, we can say:
Final Answer: The quotient is 3x with a remainder of (5x + 14)/x².