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COEFFICIENT QUESTIONS AND (1+x)^n
The coefficient of x^k in (1+x)^n is nCk. In (2-3x)^n, the term nCr 2^{n-r} (-3x)^r has coefficient nCr 2^{n-r} (-3)^r . Constant term: the power of x is 0. Independent of x: same. Meaning: peel the number in front of the wanted power, including signs from (-3)^r.
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COEFFICIENT QUESTIONS AND (1+x)^n
What is it?
The coefficient of x^k in (1+x)^n is nCk. In (2-3x)^n, the term nCr 2^{n-r} (-3x)^r has coefficient nCr 2^{n-r} (-3)^r . Constant term: the power of x is 0. Independent of x: same. Meaning: peel the number in front of the wanted power, including signs from (-3)^r.
In simple words
- The coefficient of x^k in (1+x)^n is nCk.
- In (2-3x)n, the term nCr 2n-r (-3x)r has coefficient nCr 2n-r (-3)r .
- Constant term: the power of x is 0.
- (-x)^r includes (-1)^r. For r=3 the coefficient is negative.
Important terms
COEFFICIENT QUESTIONS AND (1+x)^n: The coefficient of x^k in (1+x)^n is nCk. In (2-3x)^n, the term nCr 2^{n-r} (-3x)^r has coefficient nCr 2^{n-r} (-3)^r . Constant term: the power of x is 0. Independent of x: same. Meaning: peel the number in front of the wanted power, including signs from (-3)^r.
1. (1+x)^n: Coeff of x^3 in (1+x)^8 is 8C3=56. Sum of coefficients: put x=1, get 2^n. Alternating: x=-1, get 0 if n>0.
2. (a+bx)n: Write Tr+1 fully. Set power of x to the target. Include an-r and br. Fractions: (1+x/2)n still nCr (12)^r .
What you should learn
- State the main ideas and notation used in coefficient questions and (1+x)^n.
- Apply the rules of coefficient questions and (1+x)^n in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- 2^n sum of coeffs
- nCk wanted power
- x=-1 alternating sum
(1+x)^n
Coeff of x^3 in (1+x)^8 is 8C3=56. Sum of coefficients: put x=1, get 2^n. Alternating: x=-1, get 0 if n>0.
(a+bx)^n
Write Tr+1 fully. Set power of x to the target. Include an-r and br. Fractions: (1+x/2)n still nCr (12)^r .
Recommended for Class 11
Full school-level explanation
The coefficient of x^k in (1+x)^n is nCk. In (2-3x)^n, the term nCr 2^{n-r} (-3x)^r has coefficient nCr 2^{n-r} (-3)^r . Constant term: the power of x is 0. Independent of x: same. Meaning: peel the number in front of the wanted power, including signs from (-3)^r.
Simple binomial, then a scaled one:
WHAT — COEFFICIENT QUESTIONS AND (1+x)^n
The coefficient of x^k in (1+x)^n is nCk. In (2-3x)^n, the term nCr 2^{n-r} (-3x)^r has coefficient nCr 2^{n-r} (-3)^r . Constant term: the power of x is 0. Independent of x: same. Meaning: peel the number in front of the wanted power, including signs from (-3)^r.
RULE — 1. (1+x)^n
Coeff of x^3 in (1+x)^8 is 8C3=56. Sum of coefficients: put x=1, get 2^n. Alternating: x=-1, get 0 if n>0.
METHOD — 2. (a+bx)^n
Write Tr+1 fully. Set power of x to the target. Include an-r and br. Fractions: (1+x/2)n still nCr (12)^r .
WORKED EXAMPLE
Worked: coeff of x2 in (2-x)5. Tr+1=5Cr 25-r (-x)r want r=2: 5C2 2^3 (-1)2 = 10 8 1 = 80
(1+x)^n
Coeff of x^3 in (1+x)^8 is 8C3=56. Sum of coefficients: put x=1, get 2^n. Alternating: x=-1, get 0 if n>0.
Remember
- x=1 and x=-1 tricks
Examples
- 2^n sum of coeffs
- nCk wanted power
- x=-1 alternating sum
(a+bx)^n
Write Tr+1 fully. Set power of x to the target. Include an-r and br. Fractions: (1+x/2)n still nCr (12)^r .
Remember
- full general term
Examples
- Sign in b^r
- Power of x = r if a has no x
- Constant: r=0 if b carries all x
How to answer a coefficient questions and (1+x)^n question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- (1+x)^n : coeff of x^k is nCk
- do not drop (-3)^r
- 80
- (-1)^r
- x=1 and x=-1 tricks
- full general term
- (1+x)^n : coeff of x^k is nCk
- do not drop (-3)^r
How the ideas connect
Worked: coeff of x2 in (2-x)5. Tr+1=5Cr 25-r (-x)r want r=2: 5C2 2^3 (-1)2 = 10 8 1 = 80
Deeper understanding
Go beyond memorising COEFFICIENT QUESTIONS AND (1+x)^n. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: coeff of x2 in (2-x)5. Tr+1=5Cr 25-r (-x)r want r=2: 5C2 2^3 (-1)2 = 10 8 1 = 80
Common mistakes
Exam focus
Core Idea
1. (1+x)^n
2. (a+bx)^n
Worked Example
Exam focus
Quick revision
- (1+x)^n : coeff of x^k is nCk
- do not drop (-3)^r
- 80
- (-1)^r
- COEFFICIENT QUESTIONS AND (1+x)n: The coefficient of xk in (1+x)n is nCk. In (2-3x)n, the term nCr 2n-r (-3x)r has coefficient
- 1. (1+x)^n: Coeff of x^3 in (1+x)^8 is 8C3=56. Sum of coefficients: put x=1, get 2^n. Alternating: x=-1, get 0 i
(-x)^r includes (-1)^r. For r=3 the coefficient is negative.
For long answers on COEFFICIENT QUESTIONS AND (1+x)^n: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- (1+x)^n : coeff of x^k is nCk
- do not drop (-3)^r
- 80
- (-1)^r
Important formulas
- (1+x)^n : coeff of x^k is nCk
- do not drop (-3)^r
- 80
Common exam traps
- (-x)^r includes (-1)^r. For r=3 the coefficient is negative.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
(-x)^r includes (-1)^r. For r=3 the coefficient is negative.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
(-x)^r includes (-1)^r. For r=3 the coefficient is negative.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
(1+x)^n : coeff of x^k is nCk
do not drop (-3)^r
80
(-1)^r
The coefficient of xk in (1+x)n is nCk. In (2-3x)n, the term nCr 2n-r (-3x)r has coefficient nCr 2n-r (-3)r . Constant term: the power of x is 0. Ind
x=1 and x=-1 tricks
- Definition
- The coefficient of xk in (1+x)n is nCk. In (2-3x)n, the term nCr 2n-r (-3x)r has coefficient nCr 2n-r (-3)r . Constant term: the power of x is 0. Ind
- Formula
- x=1 and x=-1 tricks
- Example
- 2^n sum of coeffs
- Critical warning
- (-x)^r includes (-1)^r. For r=3 the coefficient is negative.
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Exam Practice
Define
Q1. Define COEFFICIENT QUESTIONS AND (1+x)^n.
Explain
Q2. Explain: (1+x)^n.
Example
Q3. Give one clear example of COEFFICIENT QUESTIONS AND (1+x)^n and say why it fits.
Mistake
Q4. State one common mistake students make about COEFFICIENT QUESTIONS AND (1+x)^n.
Exam note
Q5. Write a short exam note on COEFFICIENT QUESTIONS AND (1+x)^n: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.