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What is the Term Independent of x in Binomial Expansion?
Grade Level:
Class 8
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The 'term independent of x' in a binomial expansion is the term that does not contain the variable 'x'. It's like finding the constant value in a long mathematical expression, a number without any 'x' attached to it.
Simple Example
Quick Example
Imagine you have a recipe for chai, and the amount of sugar depends on 'x' cups of water. If you want to know how much cardamom to add, which is a fixed amount regardless of 'x', that cardamom quantity is 'independent of x'. In math, it's finding the part of an expanded expression that is just a number.
Worked Example
Step-by-Step
Let's find the term independent of x in the expansion of (x^2 + 1/x)^9.
---Step 1: Write the general term (T_r+1) for a binomial expansion (a+b)^n, which is T_r+1 = C(n, r) * a^(n-r) * b^r.
---Step 2: Here, a = x^2, b = 1/x = x^(-1), and n = 9. So, T_r+1 = C(9, r) * (x^2)^(9-r) * (x^(-1))^r.
---Step 3: Simplify the powers of x: T_r+1 = C(9, r) * x^(18-2r) * x^(-r).
---Step 4: Combine the powers of x: T_r+1 = C(9, r) * x^(18-2r-r) = C(9, r) * x^(18-3r).
---Step 5: For the term to be independent of x, the power of x must be 0. So, set 18 - 3r = 0.
---Step 6: Solve for r: 3r = 18, which means r = 6.
---Step 7: Substitute r = 6 back into the general term, but only for the coefficient part: Term = C(9, 6).
---Step 8: Calculate C(9, 6) = 9! / (6! * (9-6)!) = 9! / (6! * 3!) = (9 * 8 * 7) / (3 * 2 * 1) = 3 * 4 * 7 = 84.
ANSWER: The term independent of x is 84.
Why It Matters
Understanding terms independent of x helps in many fields! In Data Science, it can be like finding a baseline value in a dataset that doesn't change with a specific input. Engineers use this concept when designing systems where certain parameters must remain constant, regardless of varying conditions, ensuring stability and predictability in their designs.
Common Mistakes
MISTAKE: Forgetting to set the power of x to zero after combining terms. | CORRECTION: Always remember that 'independent of x' means the x-variable disappears, so its exponent must be zero.
MISTAKE: Incorrectly applying exponent rules, especially with negative exponents or powers of powers (e.g., (x^2)^3 is x^6, not x^5). | CORRECTION: Carefully review and apply exponent rules like (a^m)^n = a^(m*n) and a^m * a^n = a^(m+n).
MISTAKE: Calculating the binomial coefficient C(n, r) incorrectly. | CORRECTION: Use the formula C(n, r) = n! / (r! * (n-r)!) and simplify carefully. Remember that C(n, r) = C(n, n-r) can sometimes make calculations easier.
Practice Questions
Try It Yourself
QUESTION: Find the term independent of x in the expansion of (x + 1/x)^4. | ANSWER: 6
QUESTION: Determine the term independent of x in the expansion of (2x^2 - 1/x)^6. | ANSWER: 240
QUESTION: In the expansion of (sqrt(x) + 1/(x^2))^15, find the term independent of x. | ANSWER: C(15, 5) = 3003
MCQ
Quick Quiz
What is the first step to find the term independent of x in a binomial expansion?
Set the power of x to 1
Write down the general term of the expansion
Calculate the factorial of n
Substitute x = 0 into the expression
The Correct Answer Is:
B
The first step is always to write the general term (T_r+1) for the binomial expansion. This term will contain 'x' with a certain power, which you then set to zero to find 'r'.
Real World Connection
In the Real World
Imagine a drone delivery service like 'Dunzo' or 'Zepto'. The delivery time might depend on traffic ('x'), but the cost of the drone's battery is a fixed expense, independent of 'x'. In advanced physics, when modeling the path of a satellite, certain forces are constant regardless of its position, acting as 'terms independent of x' in complex equations.
Key Vocabulary
Key Terms
BINOMIAL EXPANSION: The process of expanding an expression like (a+b)^n into a sum of terms. | GENERAL TERM: A formula that describes any term in a sequence or series, often denoted as T_r+1. | COEFFICIENT: The numerical part of a term in an algebraic expression (e.g., 5 is the coefficient in 5x^2). | INDEPENDENT OF X: A term that does not contain the variable 'x'.
What's Next
What to Learn Next
Great job understanding terms independent of x! Next, you can explore finding specific terms in a binomial expansion, like the middle term or a term with a given power of x. This will build on your current knowledge and help you master binomial theorem applications.
