To find the remainder when the polynomial x³ + 4x² + 12x + 9 is divided by x², we can employ the polynomial long division method.
1. **Setup the Division**: Start by setting up the polynomial division. We want to divide x³ + 4x² + 12x + 9 by x².
2. **Perform the Division**: When dividing x³ by x², we get x. Now, multiply x by x² and subtract it from the original polynomial:
x³ + 4x² + 12x + 9
- (x³)
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4x² + 12x + 9
3. **Repeat the Process**: Now, divide 4x² by x², which gives you 4. Multiply 4 by x² and subtract again:
4x² + 12x + 9
- (4x²)
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12x + 9
4. **Identify the Remainder**: The next step is to note that the term 12x cannot be divided by x². Thus, the terms we are left with: 12x + 9 is the remainder since both terms are of lower degree than x².
5. **Conclusion**: Therefore, the remainder when dividing x³ + 4x² + 12x + 9 by x² is:
Remainder: 12x + 9
This remainder holds true regardless of the value of x, as long as we are dealing with polynomial division.