Summary of "AS level Maths - Binomial Theorem - (Part 7)"

Summary of “AS level Maths - Binomial Theorem - (Part 7)”

This video focuses on an advanced concept related to the binomial theorem: finding the coefficient of ( x ) (or other powers of ( x )) when ( x ) appears in both terms of a binomial expansion. It builds upon previous lessons where the coefficient was found when ( x ) was only in the first or second term.

Main Ideas and Concepts

Objective: Learn how to find the coefficient of ( x^k ) (for any power ( k )) in binomial expansions where ( x ) is present in both terms.
Key technique: Instead of expanding the entire binomial expression, use algebraic manipulation and the rules of indices (exponents) to find the value of ( r ) that corresponds to the term containing ( x^k ).
Process overview:
- Write the general term of the binomial expansion using ( r ) (the term index).
- Focus only on the powers of ( x ) in the term, ignoring coefficients and constants temporarily.
- Set the expression for the power of ( x ) equal to the desired power (e.g., 1 for ( x ), 0 for term independent of ( x )).
- Solve the resulting equation to find ( r ).
- Substitute ( r ) back into the binomial coefficient and remaining constants to find the coefficient of the term.

Detailed Methodology / Instructions

Identify the general term of the binomial expansion:

[ \binom{n}{r} ( \text{first term} )^{n-r} ( \text{second term} )^r ]

Express powers of ( x ) explicitly:
- Write the powers of ( x ) from each term separately.
- For example, if the first term is ( x ) raised to some power and the second term is ( \frac{1}{x^m} ), rewrite the second term as ( x^{-m} ).
Ignore coefficients and constants for now:
- Temporarily remove the binomial coefficient and constants that do not affect the power of ( x ).
Set up the equation for powers of ( x ):
- Add the powers of ( x ) from both terms.
- Set the sum equal to the desired power of ( x ) (e.g., 1 for coefficient of ( x ), 0 for term independent of ( x )).
Solve for ( r ):
- Solve the resulting linear equation for ( r ).
Substitute ( r ) back into the general term:
- Calculate the binomial coefficient ( \binom{n}{r} ).
- Include constants and powers of numbers.
- Simplify powers of ( x ) to confirm they reduce to the desired power.
Extract the coefficient:
- The remaining numerical factor after simplification is the coefficient of the term.

Examples Covered

Example 1: Find coefficient of ( x ) in (\left( x - \frac{3}{x} \right)^5)

General term:

[ \binom{5}{r} x^{5-r} \left(-\frac{3}{x}\right)^r ]

Powers of ( x ): (5 - r) from the first term, and (-r) from the second term (since ( \frac{1}{x} = x^{-1} ))
Equation:

[ 5 - r - r = 1 \implies 5 - 2r = 1 \implies r = 2 ]

Coefficient:

[ \binom{5}{2} \times (-3)^2 = 10 \times 9 = 90 ]

Final term:

[ 90x ]

Example 2: Find the term independent of ( x ) in (\left( 6x^2 + \frac{1}{2x} \right)^6)

General term:

[ \binom{6}{r} (6x^2)^{6-r} \left(\frac{1}{2x}\right)^r ]

Powers of ( x ): (2(6-r)) from first term, (-r) from second term
Equation:

[ 12 - 2r - r = 0 \implies 12 - 3r = 0 \implies r = 4 ]

Coefficient:

[ \binom{6}{4} \times 6^{2} \times \frac{1}{2^4} = 15 \times 36 \times \frac{1}{16} = \frac{540}{16} = \frac{135}{4} ]

Term independent of ( x ) is

[ \frac{135}{4} ]

Additional Notes

The video emphasizes the importance of understanding the manipulation of powers of ( x ) and the role of ( r ) in locating the correct term.
The method avoids expanding the entire binomial expression, saving time and effort.
The next video will cover how to calculate binomial coefficients ( \binom{n}{r} ) without a calculator, especially when ( n ) is unknown.

Speakers / Sources

Primary Speaker: The instructor presenting the lesson (name not provided).
No other speakers or external sources are mentioned.

This summary captures the core lesson and methodology for finding coefficients in binomial expansions where ( x ) appears in both terms, supported by step-by-step examples.