Binomial expansions questions with answers

Questions

1. Question 1First three terms of (1 + bx)ⁿ

   Find the first three terms in the expansion of $(1 + 3x)^{6}$ in ascending powers of $x$.

   Show the worked solution

   1. Step 1: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + …: write the terms out$(1 + 3x)^{6} = {}^{6}C_{0}(1)^{6} + {}^{6}C_{1}(1)^{5}(3x) + {}^{6}C_{2}(1)^{4}(3x)^{2} + \ldots$
   2. Step 2: Work out each coefficient: ⁿCᵣ × (the first term)ⁿ⁻ʳ × (the number with x)ʳTerm in $x^{0}$: ${}^{6}C_{0}(1)^{6}(3)^{0} = 1$Term in $x^{1}$: ${}^{6}C_{1}(1)^{5}(3)^{1} = 18$Term in $x^{2}$: ${}^{6}C_{2}(1)^{4}(3)^{2} = 135$
   3. Step 3: Write the terms together$(1 + 3x)^{6} = 1 + 18x + 135x^2 + \ldots$
2. Question 2First three terms of (1 + bx)ⁿ

   Find the first three terms in the expansion of $(1 + 3x)^{4}$ in ascending powers of $x$.

   Show the worked solution

   1. Step 1: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + …: write the terms out$(1 + 3x)^{4} = {}^{4}C_{0}(1)^{4} + {}^{4}C_{1}(1)^{3}(3x) + {}^{4}C_{2}(1)^{2}(3x)^{2} + \ldots$
   2. Step 2: Work out each coefficient: ⁿCᵣ × (the first term)ⁿ⁻ʳ × (the number with x)ʳTerm in $x^{0}$: ${}^{4}C_{0}(1)^{4}(3)^{0} = 1$Term in $x^{1}$: ${}^{4}C_{1}(1)^{3}(3)^{1} = 12$Term in $x^{2}$: ${}^{4}C_{2}(1)^{2}(3)^{2} = 54$
   3. Step 3: Write the terms together$(1 + 3x)^{4} = 1 + 12x + 54x^2 + \ldots$
3. Question 3First three terms of (a + bx)ⁿ

   Find the first three terms in the expansion of $(2 + 5x)^{5}$ in ascending powers of $x$.

   Show the worked solution

   1. Step 1: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + …: write the terms out$(2 + 5x)^{5} = {}^{5}C_{0}(2)^{5} + {}^{5}C_{1}(2)^{4}(5x) + {}^{5}C_{2}(2)^{3}(5x)^{2} + \ldots$
   2. Step 2: Work out each coefficient: ⁿCᵣ × (the first term)ⁿ⁻ʳ × (the number with x)ʳTerm in $x^{0}$: ${}^{5}C_{0}(2)^{5}(5)^{0} = 32$Term in $x^{1}$: ${}^{5}C_{1}(2)^{4}(5)^{1} = 400$Term in $x^{2}$: ${}^{5}C_{2}(2)^{3}(5)^{2} = 2000$
   3. Step 3: Write the terms together$(2 + 5x)^{5} = 32 + 400x + 2000x^2 + \ldots$
4. Question 4First three terms of (a + bx)ⁿ

   Find the first three terms in the expansion of $(2 + 5x)^{4}$ in ascending powers of $x$.

   Show the worked solution

   1. Step 1: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + …: write the terms out$(2 + 5x)^{4} = {}^{4}C_{0}(2)^{4} + {}^{4}C_{1}(2)^{3}(5x) + {}^{4}C_{2}(2)^{2}(5x)^{2} + \ldots$
   2. Step 2: Work out each coefficient: ⁿCᵣ × (the first term)ⁿ⁻ʳ × (the number with x)ʳTerm in $x^{0}$: ${}^{4}C_{0}(2)^{4}(5)^{0} = 16$Term in $x^{1}$: ${}^{4}C_{1}(2)^{3}(5)^{1} = 160$Term in $x^{2}$: ${}^{4}C_{2}(2)^{2}(5)^{2} = 600$
   3. Step 3: Write the terms together$(2 + 5x)^{4} = 16 + 160x + 600x^2 + \ldots$
5. Question 5Expand fully

   Expand $(3 + x)^{3}$ in ascending powers of $x$.

   Show the worked solution

   1. Step 1: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + …: write the terms out$(3 + x)^{3} = {}^{3}C_{0}(3)^{3} + {}^{3}C_{1}(3)^{2}(x) + {}^{3}C_{2}(3)^{1}(x)^{2} + {}^{3}C_{3}(3)^{0}(x)^{3}$
   2. Step 2: Work out each coefficient: ⁿCᵣ × (the first term)ⁿ⁻ʳ × (the number with x)ʳTerm in $x^{0}$: ${}^{3}C_{0}(3)^{3}(1)^{0} = 27$Term in $x^{1}$: ${}^{3}C_{1}(3)^{2}(1)^{1} = 27$Term in $x^{2}$: ${}^{3}C_{2}(3)^{1}(1)^{2} = 9$Term in $x^{3}$: ${}^{3}C_{3}(3)^{0}(1)^{3} = 1$
   3. Step 3: Write the terms together$(3 + x)^{3} = 27 + 27x + 9x^2 + x^3$
6. Question 6Expand fully

   Expand $(3 + 3x)^{3}$ in ascending powers of $x$.

   Show the worked solution

   1. Step 1: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + …: write the terms out$(3 + 3x)^{3} = {}^{3}C_{0}(3)^{3} + {}^{3}C_{1}(3)^{2}(3x) + {}^{3}C_{2}(3)^{1}(3x)^{2} + {}^{3}C_{3}(3)^{0}(3x)^{3}$
   2. Step 2: Work out each coefficient: ⁿCᵣ × (the first term)ⁿ⁻ʳ × (the number with x)ʳTerm in $x^{0}$: ${}^{3}C_{0}(3)^{3}(3)^{0} = 27$Term in $x^{1}$: ${}^{3}C_{1}(3)^{2}(3)^{1} = 81$Term in $x^{2}$: ${}^{3}C_{2}(3)^{1}(3)^{2} = 81$Term in $x^{3}$: ${}^{3}C_{3}(3)^{0}(3)^{3} = 27$
   3. Step 3: Write the terms together$(3 + 3x)^{3} = 27 + 81x + 81x^2 + 27x^3$