19. Newton's binomial

Newton's Binomial is one of the fundamental topics in mathematics, frequently addressed in the National High School Examination (ENEM). This theorem is a powerful tool that allows the expansion of expressions of the type (a+b)^n, where 'a' and 'b' are numbers and 'n' is a natural number. The binomial was named after Sir Isaac Newton, one of the most influential scientists of all time, who made significant contributions to mathematics and physics.

To understand Newton's Binomial, it is first necessary to understand the concept of combinations. In mathematics, a combination is the selection of items from a larger set, where the order of the items does not matter. The number of combinations of 'n' items taken 'r' at a time is given by the binomial coefficient, which is calculated using the formula nCr = n! / [(n-r)!r!], where '!' denotes the factorial.

Now, let's go to the expansion of Newton's Binomial. Given the binomial (a+b)^n, its expansion is given by:

(a+b)^n = a^n + (nC1)a^(n-1)b + (nC2)a^(n-2)b^2 + ... + b^n

Note that the coefficients of each term are the binomial coefficients (nC0, nC1, nC2, ..., nCn). Furthermore, the sum of the exponents of 'a' and 'b' in each term always adds up to 'n'. This is very useful for expanding binomials quickly without having to multiply everything.

Let's consider an example: (a+b)^3. Using Newton's Binomial, we can expand this as:

(a+b)^3 = a^3 + (3C1)a^2b + (3C2)ab^2 + b^3

This simplifies to:

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

So the expansion of the binomial (a+b)^3 is a^3 + 3a^2b + 3ab^2 + b^3.

Another way to visualize Newton's Binomial is through Pascal's Triangle, a triangular arrangement of numbers where each number is the sum of the two numbers directly above it. The binomial coefficients in Newton's Binomial expansion correspond to the numbers in Pascal's Triangle.

In summary, Newton's Binomial is a powerful tool for expanding binomials. It uses the concept of combinations and is visualized by Pascal's Triangle. This topic is often covered in ENEM and a solid understanding of Newton's Binomial can help students solve complex math problems more efficiently.

Studying Newton's Binomial not only prepares students for ENEM, but also provides a solid foundation for future studies in mathematics, computer science, physics and engineering. Therefore, it is crucial that students understand and apply this concept correctly. Regular practice of problems related to Newton's Binomial can help students become familiar with the application of this theorem and improve their mathematical skills.

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