Next I’m going to explain the **theorem of the rest**, to get the rest of a polynomial division very easily. All explained step by step and with examples and exercises solved.

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## Calculate the value of an unknown coefficient

A typical exercise that is usually asked in class related to the division between the binomial (x-a) is to calculate the value of a coefficient so that the division of a polynomial between a binomial x-a so that the division is exact.

Let’s see how to do it with the Ruffini rule step by step with the following example:

Calculate the value of m so that the division is exact:

We apply the Ruffini rule taking into account that one of the coefficients is an unknown and that the last value of the last column has to be 0:

For the last column to add 0, it must be met:

Therefore, from here we clear m, whose result is:

Therefore, when m=-3, the division will be exact.

However, to get the rest of a division between the binomial (x-a), it is not necessary to do the division. We can calculate it directly by applying the theorem of the rest.

I’ll explain it to you next.

## Theorem of the rest

The theorem of the rest says so:

box]If we divide a polynomial P(x) between the binomial (x-a), the rest of the division is equal to the numeric value of the polynomial P(x) for x=a[/box].

What does this mean?

It means we can calculate the rest of a division without having to do it when we divide a polynomial by (x-a) getting the value of the polynomial for x=a P(a).

For example:

Calculate the rest of the next division:

In this case a is equal to 1:

The numeric value of the polynomial for x=1 is:

So the rest of the division is also 3:

Let’s see another example:

What will be the rest of the next division?

The rest theorem only works for the divisions between the binomial (x-a). In this case, we have x+2, so we need to put it in the form x-a, which is x-(-2). So a is equal to -2:

The numeric value of the polynomial for x=-2 is:

Therefore, the rest of the division will be equal to 70:

Let’s look at one last example:

What is the rest of the next division?

We calculate the numeric value of the polynomial for x=1:

Whose result is 0, so the rest is also 0:

Or in other words, the division is exact.

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## Exercise solved by applying the theorem of the rest

Let’s solve the example of calculating a certain value of m, so that the division is exact, applying the theorem of the rest:

For the division to be exact, the numeric value of the polynomial for x=-3 must be 0:

On the other hand, the numeric value of the polynomial for x=-3 is:

Therefore:

What operand results:

And from here we clear m: