f(x) is the dividend, d(x) is the divisor, q(x) is the quotient, and r(x) is the remainder.
Let's start by recalling the Division Algorithm.
The Division Algorithm
Let f(x) and d(x) be polynomials such that d(x) ≠0, and the degree of d(x) is less than or equal to the degree of f(x). Then, there exist unique polynomials q(x) and r(x) such that the following equations holds, where r(x)=0 or the degree of r(x) is less than the degree of d(x).
f(x)=d(x)q(x)+r(x)
If the remainder r(x) is zero, then d(x) divides evenly into f(x).
In the equation, f(x) is the dividend, d(x) is the divisor, q(x) is the quotient, and r(x) is the remainder.
f(x)_(dividend) = d(x)^(divisor) q(x)_(quotient)+r(x)^(remainder)
The Division Algorithm is also written in the following form.
f(x)/d(x)=q(x)+r(x)/d(x)
Notice that this equation is obtained from the previous one by dividing each side of that equation by d(x). In this case, again, f(x) is the dividend, d(x) is the divisor, q(x) is the quotient, and r(x) is the remainder.
f(x)/d(x)^(dividend)_(divisor) = q(x)_(quotient)+r(x)/d(x)^(remainder)_(divisor)
Now let's consider an example polynomial f(x)=x^2+3x+5. If we divide f(x) by d(x)=x+1 using the long division, the quotient will be q(x)=x+2, and the remainder will be r(x)=3. According to the Division Algorithm, this situation is represented by the following equation.
x^2+3x+5_(dividend) = (x+1)^(divisor) (x+2)_(quotient)+3^(remainder)
We can also represent the situation using the second form of the equation from the Division Algorithm.
x^2+3x+5/x+1^(dividend)_(divisor) = x+2_(quotient)+3/x+1^(remainder)_(divisor)
