Remainder Theorem – Infinity Learn

What is the remainder theorem?

Remainder Theorem – Infinity Learn:

A theorem in mathematics that states that if a number is divided by another number, the remainder is the number that is left over after the division is complete. The remainder theorem is a mathematical theorem that states that given two integers a and b, and a positive integer n, the remainder of a/b when divided by n is always equal to the remainder of (a-b) when divided by n.

Remainder Theorem Definition

The Remainder Theorem states that if a number is divided by another number, the remainder is the result of subtracting the divisor from the dividend.

Remainder Theorem Formula

The remainder theorem states that for any two positive integers a and b, there exists a unique integer q such that a = bq + r, where r is the remainder when a is divided by b. In other words, the remainder theorem states that the remainder is always unique.

Factor Theorem

The factor theorem states that if is a prime number and is a natural number, then .

The Factor Theorem and How to Apply It

• The Factor Theorem states that a polynomial can be factored into a product of linear factors and irreducible quadratic factors. The product of the linear factors is the leading coefficient of the polynomial, and the product of the irreducible quadratic factors is the constant term.
• To apply the Factor Theorem, first find the factors of the polynomial. Second, factor the polynomial into a product of linear factors and irreducible quadratic factors. Finally, identify the leading coefficient of the product of linear factors and the constant term of the product of irreducible quadratic factors.

Remainder Theorem Proof

Proof.

We will use the following theorem:

Theorem: For any real number a and positive integer n, there exists a real number x such that a = xn + r, where r is the remainder when a is divided by n.

We will also use the following definition:

Definition: The least common multiple (lcm) of two or more numbers is the smallest number that is divisible by each of the numbers.

We will also use the following fact:

Fact: If a and b are two real numbers, then a + b is also a real number.

Proof of Theorem:

We will use the following theorem:

Theorem: For any real number a and positive integer n, there exists a real number x such that a = xn + r, where r is the remainder when a is divided by n.

Proof:

We will use the following definition:

Definition: The least common multiple (lcm) of two or more numbers is the smallest number that is divisible by each of the numbers.

We will also use the following fact:

Fact: If a and b are two real numbers, then a + b is also a real number.

Let a = xn + r, where r is the remainder when a is divided by n.

We will use the least common multiple (lcm) of

The Steps Involved in Dividing a Polynomial by a Non-Zero Polynomial

There are a few steps involved in dividing a polynomial by a non-zero polynomial.

The first step is to divide the coefficients of the polynomial to be divided by the coefficients of the divisor polynomial.

The second step is to divide the terms of the polynomial to be divided by the terms of the divisor polynomial.

The third step is to simplify the resulting polynomial.

Working of Remainder Theorem

Let us take an example to understand the working of Remainder Theorem.

Consider the number 15 ÷ 3. The remainder is 2.

This means that when 15 is divided by 3, the remainder is 2.

The remainder theorem can be used to find the remainder when a number is divided by a smaller number.

In the example above, if we want to find the remainder when 15 is divided by 2, we would use the same formula:

15 ÷ 2 = 7

The remainder is 1.

Remainder Theorem Examples

Example 1

Use the remainder theorem to find the remainder when 9 is divided by 3.

The remainder when 9 is divided by 3 is 2.