Here's an example of algebraic division:
(x3+3x2+5x+3)➗(x2+2x+3).
This is written as in arithmetic long division:
QUOTIENT
DIVISOR ) DIVIDEND
The quotient is calculated term by term.
The terms in the divisor and dividend should each be written so that the powers of x in each are in descending order. We take the first term of the dividend and divide by the first term of the divisor. So we have x3/x2=x, in other words x times x2 = x3. Therefore x is the first term of the quotient. Multiply the divisor by x: x3+2x2+3x and subtract it from the dividend. The term with the highest power of x in the dividend cancels out, leaving just two terms in the remainder. Because the divisor contains three terms we bring down the next (last, in this example) term in the dividend. We take the first term of this as a new dividend, so we divide by the first term of the divisor. So now we have x2/x2=1, in other words, 1 times x2 = x2. Now add this 1 to the quotient. Subtract 1 times the divisor from x2+2x+3, leaving zero (in this case) as the remainder, which marks the end of the division (exact division).
x + 1
x2+2x+3 ) x3+3x2+5x+3
- x3+2x2+3x ꜜ
x2+2x+3
- x2+2x+3
0
This is the general way to do algebraic long division. Not all such divisions will give a zero remainder. The division comes to an end when we get a zero remainder or when we reach the constant term in the quotient and the remainder is still not zero.