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Polynomial division


Let's think of a division of natural numbers. Share 7 out of 5 means get the quotient 1 and the rest 2. We can write:

Now let's think about the division of the polynomial A (x) by the nonzero polynomial B (x)which generates the quotient Q (x) and the rest R (x).

In this division:

  • A (x) it is the dividend;
  • B (x) is the divisor;
  • Q (x) is the quotient;
  • R (x) is the rest of the division.

The degree of R (x) must be less than the degree of B (x) or R (x) = 0.

When A (x) is divisible by B (x), we say the division is exact, that is, R (x) = 0.

Example 1

Determine the quotient of :

Resolution

  • We divide the highest degree dividend term by the highest divisor term. The result will be a quotient term:

  • We multiply per B (x) and subtract the product from A (x), getting the first partial rest:

  • We divide the highest degree term of the first partial remainder by the highest degree term of the divisor, and we get as a result a quotient term:

  • We multiply -2x per B (x) and subtract the product from the first partial remainder, obtaining the second partial remainder:

  • We divide the highest degree term of the second partial remainder by the highest degree term of the divisor, and we get as a result a quotient term:

  • We multiply 1 per B (x) and subtract the product from the second partial remainder:

Since the degree of rest is less than the degree of divisor, the division is closed.

We verified that:

Example 2

Determine the quotient of :

Resolution

We easily verify that:

In these two examples, we use the key method to effect the division of polynomials.

By the examples we find that:

quotient degree = dividend degree - divider degree

Next: Rest of the Theorem and D'Alembert's Theorem