To find the quotient when dividing the polynomial x³ + 5x² + 2x + 5 by x², we start by performing polynomial long division.
1. **Set up the division**: We divide x³ by x², which gives us x. This is the first term of our quotient.
2. **Multiply and subtract**: Multiply x by x² to get x³. Subtract this from the original polynomial:
x³ + 5x² + 2x + 5
- (x³)
-----------------
5x² + 2x + 5
3. **Repeat the process**: Now, take the new polynomial, 5x² + 2x + 5, and divide 5x² by x². This gives us 5, which is the next term of our quotient.
4. **Multiply and subtract again**: Multiply 5 by x² to get 5x² and subtract:
5x² + 2x + 5
- (5x²)
-----------------
2x + 5
5. **Final Division Step**: Now we are left with 2x + 5. Since 2x cannot be divided by x² (as the degree of 2x is less than that of x²), we can stop here.
The quotient is therefore x + 5, and the remainder is 2x + 5. Thus, when x³ + 5x² + 2x + 5 is divided by x², the final result can be expressed as:
Quotient: x + 5
Remainder: 2x + 5