What is the remainder when you divide 4x³ + 5x² + 3x + 1 by x²?

To find the remainder when dividing a polynomial by another polynomial, we can apply polynomial long division or use the Remainder Theorem. In this case, we want to divide the polynomial 4x³ + 5x² + 3x + 1 by x².

When dividing by x², the remainder will be a polynomial of degree less than 2. Therefore, we can express the remainder in the form Ax + B, where A and B are constants.

Let’s perform the polynomial long division step by step:

1. Divide the leading term of the dividend (4x³) by the leading term of the divisor (x²), which gives us 4x.
2. Multiply the entire divisor (x²) by this result (4x), yielding 4x³.
3. Subtract 4x³ from the original polynomial, which cancels out the 4x³ term:
• (4x³ + 5x² + 3x + 1) – 4x³ = 5x² + 3x + 1.
4. Now repeat the process with the new polynomial 5x² + 3x + 1.
5. Divide the leading term (5x²) by the leading term of the divisor (x²), giving us 5.
6. Multiply the divisor (x²) by this result (5), which yields 5x².
7. Subtract 5x² from 5x² + 3x + 1:
• (5x² + 3x + 1) – 5x² = 3x + 1.
8. Since the degree of 3x + 1 (degree 1) is less than 2, we stop here. Thus the remainder is 3x + 1.

In conclusion, the remainder when dividing 4x³ + 5x² + 3x + 1 by x² is 3x + 1.