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:

theorem of the rest

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:

theorem of the rest examples

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

theorem of the rest exercises

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

theorem of the rest step by step

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:

how to do the rest theorem

In this case a is equal to 1:

theorem of the rest exercises resolved

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

rest theorem exercises

So the rest of the division is also 3:

theorem of the rest definition

Let’s see another example:

What will be the rest of the next division?

uses the theorem of the rest to calculate the numerical value of

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:

What the rest theorem is for

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

ejercicios resueltos teorema del resto

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

division of polynomials theorem of the rest

Let’s look at one last example:

What is the rest of the next division?

theorem of the rest polynomials

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

teorema del resto explicación
Whose result is 0, so the rest is also 0:

find the rest of a division of polynomials

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:

calculate the rest of a polynomial division without doing it

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

resolved theorem examples of the rest

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

theorem of the rest solved examples

Therefore:

theorem of the rest find m

What operand results:

rest theorem as applied

And from here we clear m:

exercises applying theorem of the rest