A term or coefficient questions with answers

Questions

1. Question 1Coefficient of xʳ

   Find the coefficient of $x^{3}$ in the expansion of $(1 + 4x)^{8}$.

   Show the worked solution

   1. Step 1: Write the general term: Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ (bx)ʳ$T_{r+1} = {}^{8}C_{r}(1)^{8 - r}(4x)^{r}$
   2. Step 2: The power of x is r: match it to the power of x you want$r = 3$
   3. Step 3: Substitute your r, then work out each part$T_{4} = {}^{8}C_{3}(1)^{5}(4x)^{3}$$= 56 \times 1 \times 64x^{3}$$\text{Coefficient} = 3584$
2. Question 2Coefficient of xʳ

   Find the coefficient of $x^{2}$ in the expansion of $(1 - 3x)^{5}$.

   Show the worked solution

   1. Step 1: Write the general term: Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ (bx)ʳ$T_{r+1} = {}^{5}C_{r}(1)^{5 - r}(-3x)^{r}$
   2. Step 2: The power of x is r: match it to the power of x you want$r = 2$
   3. Step 3: Substitute your r, then work out each part$T_{3} = {}^{5}C_{2}(1)^{3}(-3x)^{2}$$= 10 \times 1 \times 9x^{2}$$\text{Coefficient} = 90$
3. Question 3Term independent of x

   Find the term independent of $x$ in the expansion of $\left(x + \dfrac{3}{x}\right)^{8}$.

   Show the worked solution

   1. Step 1: Write the general term: Tᵣ₊₁ = ⁿCᵣ (first term)ⁿ⁻ʳ (second term)ʳ, then collect the powers of x$T_{r+1} = {}^{8}C_{r}(x)^{8 - r}\left(\dfrac{3}{x}\right)^{r}$$= {}^{8}C_{r}(3)^{r} x^{8 - 2r}$
   2. Step 2: Independent of x: the power of x is 0$8 - 2r = 0$$r = 4$
   3. Step 3: Substitute your r, then work out each part$T_{5} = {}^{8}C_{4}(3)^{4}$$= 70 \times 81$$= 5670$
4. Question 4Term independent of x

   Find the term independent of $x$ in the expansion of $\left(x + \dfrac{3}{x}\right)^{4}$.

   Show the worked solution

   1. Step 1: Write the general term: Tᵣ₊₁ = ⁿCᵣ (first term)ⁿ⁻ʳ (second term)ʳ, then collect the powers of x$T_{r+1} = {}^{4}C_{r}(x)^{4 - r}\left(\dfrac{3}{x}\right)^{r}$$= {}^{4}C_{r}(3)^{r} x^{4 - 2r}$
   2. Step 2: Independent of x: the power of x is 0$4 - 2r = 0$$r = 2$
   3. Step 3: Substitute your r, then work out each part$T_{3} = {}^{4}C_{2}(3)^{2}$$= 6 \times 9$$= 54$
5. Question 5In a product

   Find the coefficient of $x$ in the expansion of $(1 - 3x)(1 + 3x)^{6}$.

   Show the worked solution

   1. Step 1: Expand (1 + 3x)ⁿ as far as the x term$(1 + 3x)^{6} = 1 + {}^{6}C_{1}(3x) + {}^{6}C_{2}(3x)^{2} + \ldots$$= 1 + 18x + 135x^2 + \ldots$
   2. Step 2: Only two products give x: (the number in front) × (x term), and (the x term in front) × (x⁰ term)$\text{Coefficient} = 1(18) + (-3)(1)$$= 15$
6. Question 6In a product

   Find the coefficient of $x$ in the expansion of $(1 + 4x)(1 + 2x)^{4}$.

   Show the worked solution

   1. Step 1: Expand (1 + 2x)ⁿ as far as the x term$(1 + 2x)^{4} = 1 + {}^{4}C_{1}(2x) + {}^{4}C_{2}(2x)^{2} + \ldots$$= 1 + 8x + 24x^2 + \ldots$
   2. Step 2: Only two products give x: (the number in front) × (x term), and (the x term in front) × (x⁰ term)$\text{Coefficient} = 1(8) + 4(1)$$= 12$