## Remainder Theorem for Polynomials

### The Remainder Theorem is a result in algebra that relates to polynomial division

The Remainder Theorem is a result in algebra that relates to polynomial division. It allows us to find the remainder of a polynomial division when the dividend (the polynomial being divided) is divided by a linear binomial of the form (x – a), where ‘a’ is a constant.

The theorem states that if a polynomial P(x) is divided by (x – a), then the remainder obtained is equal to P(a). In other words, if we substitute the value ‘a’ into the polynomial P(x), the output will be the remainder of the division.

In symbolic form, the remainder theorem can be written as:

If P(x) is divided by (x – a), then the remainder R(x) satisfies:

P(x) = (x – a) * Q(x) + R(x)

Where:

P(x) is the polynomial being divided,

(x – a) is the linear binomial divisor,

Q(x) is the quotient obtained from the division, and

R(x) is the remainder.

To use the remainder theorem to find the remainder of a polynomial division, follow these steps:

1. Write the polynomial P(x) in descending order of degree.

2. Set up the division with the divisor (x – a) and divide normally.

3. After performing the division, the remainder obtained will be the value of P(a).

For example, let’s say we have the polynomial P(x) = 3x^3 – 5x^2 + 2x + 4, and we want to divide it by (x – 2) to find the remainder. Using the remainder theorem, we substitute ‘a = 2’ into P(x):

P(2) = 3(2)^3 – 5(2)^2 + 2(2) + 4

= 3(8) – 5(4) + 4 + 4

= 24 – 20 + 4 + 4

= 12

Therefore, the remainder of the division P(x) ÷ (x – 2) is 12.

The remainder theorem is useful in various applications of polynomial division, such as finding factors and roots of polynomials, evaluating polynomial functions, and solving polynomial equations.

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