In every division we have dividend, divisor, quotient and remainder, as we are talking about division of polynomial by polynomial, we will have:
To dividend a polynomial G(x)
To divider a polynomial D(x)
To quotient a polynomial Q(x)
To rest (can be zero) a polynomial R(x)
Actual proof:
There are some observations to be made, such as:
- at the end of the division, the remainder must always be smaller than the divisor: R(x) < D(x).
- when the remainder equals zero, the division is considered exact, that is, the dividend is divisible by the divisor. R(x) = 0.
Note the division of polynomial by polynomial below, let's start with an example, each step taken in the development of the division will be explained.
given the division
(12x3 + 9 - 4x): (x + 2x2 + 3)
Before starting the operation we have to make some checks:
- if all the polynomials are in order according to the powers of x.
In the case of our division, we must order, thus:
(12x3 - 4x + 9): (2x2 + x + 3)
- observe if the polynomial G(x) is not missing any term, if it is, we must complete.
In the 12x polynomial3 - 4x + 9 the x term is missing2, completing it will look like this:
12x3 + 0x2 - 4x + 9
Now we can start the division:
- G(x) has 3 terms and D(x) has 3 terms. We take the 1st term of G(x) and divide it by the 1st term of D(x): 12x3: 2x2 = 6x, the result will multiply the polynomial 2x2 + x + 3 and the result of this multiplication we will subtract by the polynomial 12x3 + 0x2 - 4x + 9. So we will have:
- R(x) > D(x), we can continue the division, repeating the same process as before. Finding now the second term of Q(x).
R(x) < D(x), we do not continue the division, concluding that:
The quotient is 6x – 3 and the rest is –19x + 18.
by Danielle de Miranda
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/divisao-polinomio-por-polinomio.htm