﻿ Theorem of the rest. What it is and how it is applied. Resolved exercises.

Theorem of the rest. What it is and how it is applied. Resolved exercises.

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.

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: