MathematicsChapter 7: Mathematical Induction and Binomial Theorem
COEFFICIENT QUESTIONS AND (1+x)^n
The coefficient of x^k in (1+x)^n is nCk.
Learning objectives
- 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.
Definition
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.
Where / symbols
- 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 T_{r+1} fully. Set power of x to the target. Include a^{n-r} and b^r. Fractions: (1+x/2)^n still nCr (1/2)^r .
Overview
Simple binomial, then a scaled one:
(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.
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 .
Examples Sign in b^r · Power of x = r if a has no x · Constant: r=0 if b carries all x
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
How to answer a coefficient questions and (1+x)^n question
- 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.
Worked example
- coeff of x^2 in (2-x)^5.
- Tr+1=5Cr 25-r (-x)r
- want r=2: 5C2 * 23 * (-1)2
- = 10 * 8 * 1 = 80
In simple words
(-x)^r includes (-1)^r. For r=3 the coefficient is negative.
Key points
- (1+x)^n : coeff of x^k is nCk
- do not drop (-3)^r
- x=1 and x=-1 tricks
- full general term
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.COEFFICIENT QUESTIONS AND (1+x)^n
Sticky Notes · Quick Revision
Core Idea
- 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.
1. (1+x)^n
- Coeff of x3 in (1+x)8 is 8C3=56.
- Sum of coefficients: put x=1, get 2n.
- Alternating: x=-1, get 0 if n>0.
2. (a+bx)^n
- Write T_{r+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 x^2 in (2-x)^5.
- Tr+1=5Cr 25-r (-x)r
- want r=2: 5C2 * 23 * (-1)2
- = 10 * 8 * 1 = 80
Common Misconception
- (-x)^r includes (-1)^r. For r=3 the coefficient is negative.
FBISE Exam Check
- State the rule for coefficient questions and (1+x)^n before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- 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.
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