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What is the Coefficient of a Term in a Binomial Expansion?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The coefficient of a term in a binomial expansion is the numerical factor that multiplies the variables in that specific term. It tells us 'how many times' that particular combination of variables appears in the expanded form of a binomial expression like (a + b)^n.
Simple Example
Quick Example
Imagine you're buying 'chai' and 'samosas'. If you buy (chai + samosa)^2, it means (chai + samosa) * (chai + samosa). When you expand this, you get chai^2 + 2*chai*samosa + samosa^2. Here, the '2' in '2*chai*samosa' is the coefficient. It shows there are two ways to get one chai and one samosa combination.
Worked Example
Step-by-Step
Let's find the coefficient of the x^2y term in the expansion of (x + y)^3.
Step 1: Write down the binomial expression and its power: (x + y)^3.
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Step 2: Expand the binomial expression. (x + y)^3 = (x + y) * (x + y) * (x + y).
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Step 3: Multiply them out: (x^2 + 2xy + y^2) * (x + y).
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Step 4: Continue multiplying: x^2(x + y) + 2xy(x + y) + y^2(x + y).
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Step 5: Simplify: x^3 + x^2y + 2x^2y + 2xy^2 + xy^2 + y^3.
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Step 6: Combine like terms: x^3 + (x^2y + 2x^2y) + (2xy^2 + xy^2) + y^3.
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Step 7: Final expanded form: x^3 + 3x^2y + 3xy^2 + y^3.
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Step 8: Identify the term x^2y. Its numerical factor is 3. So, the coefficient of the x^2y term is 3.
Why It Matters
Understanding coefficients is crucial in fields like AI/ML for calculating probabilities and in Physics for describing wave patterns. Engineers use them to model complex systems, and even in Space Technology, they help predict orbital paths, making sure our satellites like those from ISRO stay on track.
Common Mistakes
MISTAKE: Confusing the exponent with the coefficient. For example, in 3x^2, thinking 2 is the coefficient. | CORRECTION: The coefficient is the number multiplying the variable part, which is 3. The exponent (power) is 2.
MISTAKE: Forgetting that a term like x^3 actually has a coefficient. | CORRECTION: If no number is explicitly written, the coefficient is always 1. So, the coefficient of x^3 is 1.
MISTAKE: Incorrectly identifying the term. For example, in 3x^2y + 2xy^2, identifying 2 as the coefficient of x^2y. | CORRECTION: Always look at the specific variable combination requested. For x^2y, the coefficient is 3. For xy^2, the coefficient is 2.
Practice Questions
Try It Yourself
QUESTION: What is the coefficient of the 'ab' term in the expansion of (a + b)^2? | ANSWER: 2
QUESTION: Find the coefficient of the x^3 term in the expansion of (x + 2)^3. (Hint: (x+2)^3 = x^3 + 3*x^2*2 + 3*x*2^2 + 2^3) | ANSWER: 1
QUESTION: In the expansion of (2p + q)^2, what is the coefficient of the pq term? | ANSWER: 4
MCQ
Quick Quiz
What is the coefficient of the y^2 term in the expansion of (x + y)^4?
1
4
6
2
The Correct Answer Is:
C
The expansion of (x + y)^4 is x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4. The term with y^2 is 6x^2y^2, so its coefficient is 6.
Real World Connection
In the Real World
Imagine a cricket commentator calculating how many different combinations of runs a batsman can score in an over. The coefficients from binomial expansions can help figure out the number of ways to get specific outcomes, like scoring 'X' runs with 'Y' boundaries, similar to how data analysts use them in sports analytics for teams like Mumbai Indians or CSK.
Key Vocabulary
Key Terms
BINOMIAL: An algebraic expression with two terms, like (a + b) | EXPANSION: The process of multiplying out a bracketed expression to remove the brackets | TERM: A single number or variable, or numbers and variables multiplied together, in an expression | NUMERICAL FACTOR: The number part of a term that multiplies the variable part
What's Next
What to Learn Next
Great job understanding coefficients! Next, explore 'Pascal's Triangle' and the 'Binomial Theorem'. These powerful tools will show you much faster ways to find coefficients for even larger powers without expanding everything step-by-step.
