Dividing Polynomials: (x^4  6x^3 + 4x^2 + 3x + 1) / (x  3)
This article will explore the division of the polynomial (x^4  6x^3 + 4x^2 + 3x + 1) by (x  3). We'll utilize the long division method to find the quotient and remainder of this division.
Long Division of Polynomials
Long division for polynomials is similar to the long division process for numbers. Here's how it works:

Set up the problem: Write the dividend (x^4  6x^3 + 4x^2 + 3x + 1) inside the division symbol and the divisor (x  3) outside.

Focus on the leading terms: Divide the leading term of the dividend (x^4) by the leading term of the divisor (x). This gives us x^3. Write this above the division symbol, aligned with the x^3 term.

Multiply the quotient by the divisor: Multiply the quotient (x^3) by the divisor (x  3) to get (x^4  3x^3).

Subtract: Subtract the result from the dividend. This gives us (3x^3 + 4x^2 + 3x + 1).

Bring down the next term: Bring down the next term of the dividend (4x^2).

Repeat steps 25: Now we repeat the process with the new dividend (3x^3 + 4x^2 + 3x + 1). Divide the leading term (3x^3) by the leading term of the divisor (x), resulting in 3x^2. Write this above the division symbol.

Continue until the degree of the remainder is less than the degree of the divisor: Continue this process until the degree of the remainder is less than the degree of the divisor.
Here's a visual representation of the process:
x^3  3x^2 + x + 6
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x  3  x^4  6x^3 + 4x^2 + 3x + 1
(x^4  3x^3)

3x^3 + 4x^2
(3x^3 + 9x^2)

5x^2 + 3x
(5x^2 + 15x)

12x + 1
(12x + 36)

35
Therefore, the quotient is (x^3  3x^2 + x + 6) and the remainder is 35.
This can be written in the form: (x^4  6x^3 + 4x^2 + 3x + 1) / (x  3) = (x^3  3x^2 + x + 6)  35/(x  3)
This means the original polynomial can be represented as the product of (x  3) and (x^3  3x^2 + x + 6) with a remainder of 35.