Binomial Expansion for Positive Integral Exponents

Welcome to the mastery of Binomial Expansion for positive integral exponents! This powerful tool is fundamental in advanced algebra and is frequently tested in competitive exams like JEE. We'll break down the concept, its formula, and how to apply it effectively.

Understanding the Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form $(a+b)^n$ , where 'n' is a positive integer. Instead of tedious multiplication, the theorem offers a systematic way to find each term in the expansion.

The Binomial Theorem expands $(a+b)^n$ into a sum of terms.

The expansion of $(a+b)^n$ involves terms where the power of 'a' decreases from 'n' to 0, and the power of 'b' increases from 0 to 'n'. Each term is multiplied by a specific coefficient.

The general form of the binomial expansion for a positive integer 'n' is given by:

$(a+b)^n = \binom{n}{0}a^n b^0 + \binom{n}{1}a^{n-1} b^1 + \binom{n}{2}a^{n-2} b^2 + \dots + \binom{n}{r}a^{n-r} b^r + \dots + \binom{n}{n}a^0 b^n$

This can be written in summation notation as:

$(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$

Here, $\binom{n}{r}$ is the binomial coefficient, read as 'n choose r', and is calculated as $\frac{n!}{r!(n-r)!}$ .

The Binomial Coefficients

The coefficients in the binomial expansion are crucial. They are determined by binomial coefficients, often visualized using Pascal's Triangle. Each coefficient $\binom{n}{r}$ represents the number of ways to choose 'r' items from a set of 'n' items.

What is the formula for the binomial coefficient

\binom{n}{r}

$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Let's look at the first few expansions to see the pattern:

Expansion	Result
$(a+b)^1$	$a+b$
$(a+b)^2$	$a^2 + 2ab + b^2$
$(a+b)^3$	$a^3 + 3a^2b + 3ab^2 + b^3$
$(a+b)^4$	$a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$

Key Terms and Properties

Understanding the general term is vital for solving problems efficiently. The $(r+1)^{th}$ term in the expansion of $(a+b)^n$ is given by $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ .

Remember: The sum of the exponents of 'a' and 'b' in each term of the expansion of $(a+b)^n$ is always 'n'.

The binomial expansion can be visualized as a systematic process of combining terms. For $(a+b)^n$ , we are essentially choosing 'a' or 'b' from each of the 'n' factors. The binomial coefficient $\binom{n}{r}$ counts how many ways we can choose 'b' exactly 'r' times (and thus 'a' $n-r$ times) from the 'n' factors. This combinatorial interpretation is key to understanding the theorem's structure.

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Applications in Competitive Exams

In competitive exams, you'll encounter problems that require finding specific terms, coefficients, or properties of the expansion. Common tasks include:

Finding the $k^{th}$ term.
Finding the coefficient of a specific term (e.g., $x^5$ ).
Finding the constant term.
Determining the middle term(s).
Using the expansion to approximate values or solve number theory problems.

What is the

(r+1)^{th}

term in the expansion of

(a+b)^n

$T_{r+1} = \binom{n}{r} a^{n-r} b^r$

Example Problem

Find the 5th term in the expansion of $(2x - 3y)^{10}$ .

Here, $a = 2x$ , $b = -3y$ , and $n = 10$ . We need the 5th term, which means $r+1 = 5$ , so $r = 4$ .

Using the formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ :

$T_5 = \binom{10}{4} (2x)^{10-4} (-3y)^4$ $T_5 = \binom{10}{4} (2x)^6 (-3y)^4$

Calculate $\binom{10}{4} = \frac{10!}{4!6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 10 \times 3 \times 7 = 210$ .

$(2x)^6 = 2^6 x^6 = 64x^6$ $(-3y)^4 = (-3)^4 y^4 = 81y^4$

So, $T_5 = 210 \times 64x^6 \times 81y^4$ $T_5 = 1088640 x^6 y^4$

This systematic approach ensures accuracy in solving such problems.

Learning Resources

Binomial Theorem - Wikipedia(wikipedia)

Provides a comprehensive overview of the binomial theorem, its history, and generalizations.

Binomial Expansion - Khan Academy(tutorial)

Offers video lessons and practice exercises on binomial expansion, starting from the basics.

Binomial Theorem Explained - Brilliant.org(documentation)

A clear explanation of the binomial theorem with interactive examples and problem-solving strategies.

JEE Mathematics: Binomial Theorem - Vedantu(blog)

Focuses on the application of the binomial theorem for competitive exams like JEE, with solved examples.

Understanding Pascal's Triangle and Binomial Coefficients(documentation)

Explains Pascal's Triangle and its direct relationship to binomial coefficients, aiding in understanding the pattern.

Binomial Theorem Problems with Solutions - Toppr(blog)

Provides a collection of solved problems on the binomial theorem, covering various question types relevant to exams.

The Binomial Theorem - Mathematics LibreTexts(documentation)

A detailed chapter on the binomial theorem, including its proof and applications.

Binomial Expansion for Positive Integral Exponents - YouTube(video)

A video tutorial demonstrating the binomial expansion for positive integral exponents with clear examples.

NCERT Class 11 Maths Chapter 8: Binomial Theorem(documentation)

The official NCERT textbook chapter on the Binomial Theorem, offering a foundational understanding.

Binomial Theorem: Formula, Properties, and Applications(blog)

A comprehensive guide to the binomial theorem, covering its formula, properties, and common applications.