To find the quotient when dividing the polynomial x³ + 5x² + 2x + 5 by x – 2, we will use polynomial long division.
1. **Setup:** Write the polynomial you want to divide (the dividend) and the divisor. Here, we’re dividing x³ + 5x² + 2x + 5 by x – 2.
2. **First Division:** Divide the leading term of the dividend (x³) by the leading term of the divisor (x), which gives you x². Place this above the long division line.
3. **Multiply and Subtract:** Multiply x² by the entire divisor (x – 2), which results in x³ – 2x². Subtract this from the original polynomial:
- x³ + 5x² + 2x + 5
- – (x³ – 2x²)
This gives you 7x² + 2x + 5.
4. **Second Division:** Now repeat the process. Divide the leading term of your new polynomial (7x²) by the leading term of the divisor (x). This gives you 7x. Place that above the division line.
5. **Multiply and Subtract Again:** Multiply 7x by the divisor (x – 2), giving you 7x² – 14x. Subtract this from 7x² + 2x + 5:
- 7x² + 2x + 5
- – (7x² – 14x)
This simplifies to 16x + 5.
6. **Final Division:** Divide the new leading term (16x) by the leading term of the divisor (x), resulting in 16. Write 16 above the division line.
7. **Final Multiply and Subtract:** Multiply 16 by (x – 2), resulting in 16x – 32. Subtract this from 16x + 5:
- 16x + 5
- – (16x – 32)
When you do this, you obtain a remainder of 37.
8. **Result:** Therefore, the quotient when x³ + 5x² + 2x + 5 is divided by x – 2 is x² + 7x + 16 with a remainder of 37.
So, you can express the final answer as:
Quotient: x² + 7x + 16, Remainder: 37
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