Regarding polynomials – What does the remainder theorem consist of?

According to the Maths language, the remainder of the polynomial division `P(x)` by a binomial `(x-a)` equals the value of `P(a).` This confirmation is known as being the Remainder Theorem or D'Alembert Theorem, But what is it for? In short, it enables us to find the remainder of a polynomial division by another one, without being necessary to solve the division. We have to bear in mind that this theorem can only be used if the divisor appears like `x-a`, in which `a` is the constant.

• Let's us explain through an examples o as to clarify it better. Let´s suppose that we have:

  `P(x) = 2x^3-3x^2+5x+2`

  And we want to know what is the remainder of this polynomial division by the following one:

  `Q(x) = x+1`

  In this situation we do not need to solve the division. It's enough to calculate:

  `P(-1) = 2xx(-1)^3-3xx(-1)^2+5xx(-1)+2=-2-3-5+2=-8`

  Thus we can notice that the remainder is `-8` as it doesn´t result in a zero remainder, it is said that `P(x)` can be divided by `Q(x)`.

• Let´s see another example. Supposing that we have:

  `P(x) = x^4-2x^3+x-2`

  And we want to know what is the division remainder or this polynomial by the following one:

  `Q(x) = x-2`.

  Once more it is not necessary to solve the division. It´s enough to calculate:

  `P(2) = 2^4-2xx(2)^3+2-2=16-16+2-2=0`

  Thus we can notice that the remainder is `0` and so `P(x)` can be divided by `Q(x)` and number `2` is a zero or the polynomial root of `P(x)`.

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