Final answer:
To divide the given polynomial by the given divisor, we use long division. The quotient is x² + 4x + 11, with a remainder of 23.
Step-by-step explanation:
When dividing polynomials, similar to dividing numbers, we use long division. Let's divide the given polynomial (x³ + 6x² + 3x + 1) by (x - 2).
First, we divide the first term of the polynomial, x³, by the first term of the divisor, x. This gives us x². Now we multiply the entire divisor, (x - 2), by x², which gives us (x³ - 2x²), and subtract it from the original polynomial. This gives us (4x² + 3x + 1) as the new dividend.
Now we repeat the process with the new dividend and the original divisor. We divide the first term of the new dividend, 4x², by the first term of the divisor, x, resulting in 4x. We multiply the entire divisor by 4x, giving us (4x² - 8x), and subtract it from the new dividend. This leaves us with (11x + 1) as the new dividend.
Lastly, we divide the first term of the new dividend, 11x, by the first term of the divisor, x, resulting in 11. Multiplying the entire divisor by 11, we get (11x - 22), and subtract it from the new dividend. This gives us a remainder of 23.
Therefore, the quotient of the division is x² + 4x + 11, with a remainder of 23.
Learn more about Dividing polynomials