Binomial Theorem

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Answer Key & Explanations

Binomial Theorem · Grade 12 · Worksheet 1

1. Expand (x + 2y)^4 = ? Answer: x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4 Solution: Use the binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k For (x + 2y)^4, a = x, b = 2y, n = 4 The binomial coefficients for n=4 are: 1, 4, 6, 4, 1 k=0: C(4,0) * x^4 * (2y)^0 = 1 * x^4 * 1 = x^4 k=1: C(4,1) * x^3 * (2y)^1 = 4 * x^3 * 2y = 8x^3y k=2: C(4,2) * x^2 * (2y)^2 = 6 * x^2 * 4y^2 =…
   Full step-by-step solution

   Step 1: Use the binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Step 2: For (x + 2y)^4, a = x, b = 2y, n = 4 Step 3: The binomial coefficients for n=4 are: 1, 4, 6, 4, 1 Step 4: Expand each term: k=0: C(4,0) * x^4 * (2y)^0 = 1 * x^4 * 1 = x^4 k=1: C(4,1) * x^3 * (2y)^1 = 4 * x^3 * 2y = 8x^3y k=2: C(4,2) * x^2 * (2y)^2 = 6 * x^2 * 4y^2 = 24x^2y^2 k=3: C(4,3) * x^1 * (2y)^3 = 4 * x * 8y^3 = 32xy^3 k=4: C(4,4) * x^0 * (2y)^4 = 1 * 1 * 16y^4 = 16y^4 Step 5: Combine all terms: x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4 The answer is x^4 + 8x^3y + 24x^2y^2 + 32xy^3 + 16y^4.

2. Mason is a financial analyst studying the growth of an investment portfolio. The portfolio's value in thousands of dollars after t years is modeled by the expression (2 + 0.5t)^8. Using the binomial theorem, expand (2 + 0.5t)^8 completely. Write your answer as a polynomial in descending powers of t. Answer: 256 + 512t + 448t^2 + 224t^3 + 70t^4 + 14t^5 + 1.75t^6 + 0.125t^7 + 0.00390625t^8 Solution: Identify a = 2, b = 0.5t, and n = 8. Use binomial theorem: (a + b)^8 = sum_{k=0}^{8} C(8,k) * a^(8-k) * b^k. Compute binomial coefficients using Pascal's triangle row 8: 1, 8, 28, 56, 70, 56, 28, 8, 1.
   Full step-by-step solution

   Step 1: Identify a = 2, b = 0.5t, and n = 8. Step 2: Use binomial theorem: (a + b)^8 = sum_{k=0}^{8} C(8,k) * a^(8-k) * b^k. Step 3: Compute binomial coefficients using Pascal's triangle row 8: 1, 8, 28, 56, 70, 56, 28, 8, 1. Step 4: Expand each term: - k=0: 1 * 2^8 * (0.5t)^0 = 1 * 256 * 1 = 256 - k=1: 8 * 2^7 * (0.5t)^1 = 8 * 128 * 0.5t = 512t - k=2: 28 * 2^6 * (0.5t)^2 = 28 * 64 * 0.25t^2 = 28 * 16t^2 = 448t^2 - k=3: 56 * 2^5 * (0.5t)^3 = 56 * 32 * 0.125t^3 = 56 * 4t^3 = 224t^3 - k=4: 70 * 2^4 * (0.5t)^4 = 70 * 16 * 0.0625t^4 = 70 * 1t^4 = 70t^4 - k=5: 56 * 2^3 * (0.5t)^5 = 56 * 8 * 0.03125t^5 = 56 * 0.25t^5 = 14t^5 - k=6: 28 * 2^2 * (0.5t)^6 = 28 * 4 * 0.015625t^6 = 28 * 0.0625t^6 = 1.75t^6 - k=7: 8 * 2^1 * (0.5t)^7 = 8 * 2 * 0.0078125t^7 = 8 * 0.015625t^7 = 0.125t^7 - k=8: 1 * 2^0 * (0.5t)^8 = 1 * 1 * 0.00390625t^8 = 0.00390625t^8 Step 5: Write polynomial in descending powers of t: 256 + 512t + 448t^2 + 224t^3 + 70t^4 + 14t^5 + 1.75t^6 + 0.125t^7 + 0.00390625t^8 The answer is 256 + 512t + 448t^2 + 224t^3 + 70t^4 + 14t^5 + 1.75t^6 + 0.125t^7 + 0.00390625t^8.

3. Expand (4x - y)^6 = ? Answer: 4096x^6 - 6144x^5y + 3840x^4y^2 - 1280x^3y^3 + 240x^2y^4 - 24xy^5 + y^6 Solution: Use binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Here a = 4x, b = -y, n = 6 Binomial coefficients for n=6 are: 1, 6, 15, 20, 15, 6, 1 k=0: C(6,0) * (4x)^6 * (-y)^0 = 1 * 4096x^6 * 1 = 4096x^6 k=1: C(6,1) * (4x)^5 * (-y)^1 = 6 * 1024x^5 * (-y) = -6144x^5y k=2: C(6,2) * (4x)^4 * (-y)^2 =…
   Full step-by-step solution

   Step 1: Use binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Step 2: Here a = 4x, b = -y, n = 6 Step 3: Binomial coefficients for n=6 are: 1, 6, 15, 20, 15, 6, 1 Step 4: Expand each term: k=0: C(6,0) * (4x)^6 * (-y)^0 = 1 * 4096x^6 * 1 = 4096x^6 k=1: C(6,1) * (4x)^5 * (-y)^1 = 6 * 1024x^5 * (-y) = -6144x^5y k=2: C(6,2) * (4x)^4 * (-y)^2 = 15 * 256x^4 * y^2 = 3840x^4y^2 k=3: C(6,3) * (4x)^3 * (-y)^3 = 20 * 64x^3 * (-y^3) = -1280x^3y^3 k=4: C(6,4) * (4x)^2 * (-y)^4 = 15 * 16x^2 * y^4 = 240x^2y^4 k=5: C(6,5) * (4x)^1 * (-y)^5 = 6 * 4x * (-y^5) = -24xy^5 k=6: C(6,6) * (4x)^0 * (-y)^6 = 1 * 1 * y^6 = y^6 Step 5: Combine all terms: 4096x^6 - 6144x^5y + 3840x^4y^2 - 1280x^3y^3 + 240x^2y^4 - 24xy^5 + y^6

4. Expand (4x - 2y)^6 = ? Answer: 4096x^6 - 12288x^5y + 15360x^4y^2 - 10240x^3y^3 + 3840x^2y^4 - 768xy^5 + 64y^6 Solution: Use the binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Here a = 4x, b = -2y, n = 6 The binomial coefficients for n=6 are: 1, 6, 15, 20, 15, 6, 1 k=0: C(6,0) * (4x)^6 * (-2y)^0 = 1 * 4096x^6 * 1 = 4096x^6 k=1: C(6,1) * (4x)^5 * (-2y)^1 = 6 * 1024x^5 * (-2y) = -12288x^5y k=2: C(6,2) *…
   Full step-by-step solution

   Step 1: Use the binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Step 2: Here a = 4x, b = -2y, n = 6 Step 3: The binomial coefficients for n=6 are: 1, 6, 15, 20, 15, 6, 1 Step 4: Expand each term: k=0: C(6,0) * (4x)^6 * (-2y)^0 = 1 * 4096x^6 * 1 = 4096x^6 k=1: C(6,1) * (4x)^5 * (-2y)^1 = 6 * 1024x^5 * (-2y) = -12288x^5y k=2: C(6,2) * (4x)^4 * (-2y)^2 = 15 * 256x^4 * 4y^2 = 15360x^4y^2 k=3: C(6,3) * (4x)^3 * (-2y)^3 = 20 * 64x^3 * (-8y^3) = -10240x^3y^3 k=4: C(6,4) * (4x)^2 * (-2y)^4 = 15 * 16x^2 * 16y^4 = 3840x^2y^4 k=5: C(6,5) * (4x)^1 * (-2y)^5 = 6 * 4x * (-32y^5) = -768xy^5 k=6: C(6,6) * (4x)^0 * (-2y)^6 = 1 * 1 * 64y^6 = 64y^6 Step 5: Combine all terms: 4096x^6 - 12288x^5y + 15360x^4y^2 - 10240x^3y^3 + 3840x^2y^4 - 768xy^5 + 64y^6

5. Hana is a structural engineer designing a support beam for a new bridge. The beam's deflection under load is modeled by the expression (2x - 3)^6, where x represents the distance in meters from the left support. To analyze the stress distribution, Hana needs to expand this binomial expression fully. Using the binomial theorem, expand (2x - 3)^6 and simplify all terms. Answer: 64x^6 - 576x^5 + 2160x^4 - 4320x^3 + 4860x^2 - 2916x + 729 Solution: Identify the binomial theorem: (a + b)^n = sum_{k=0}^{n} C(n,k) * a^(n-k) * b^k, where C(n,k) = n! / (k! * (n-k)!).
   Full step-by-step solution

   Step 1: Identify the binomial theorem: (a + b)^n = sum_{k=0}^{n} C(n,k) * a^(n-k) * b^k, where C(n,k) = n! / (k! * (n-k)!). Here, a = 2x, b = -3, and n = 6. Step 2: List the binomial coefficients for n = 6 using Pascal's triangle or combinations: C(6,0)=1, C(6,1)=6, C(6,2)=15, C(6,3)=20, C(6,4)=15, C(6,5)=6, C(6,6)=1. Step 3: Write the expansion term by term: Term 1 (k=0): C(6,0)*(2x)^6*(-3)^0 = 1 * 64x^6 * 1 = 64x^6 Term 2 (k=1): C(6,1)*(2x)^5*(-3)^1 = 6 * 32x^5 * (-3) = 6 * 32x^5 * (-3) = 192x^5 * (-3) = -576x^5 Term 3 (k=2): C(6,2)*(2x)^4*(-3)^2 = 15 * 16x^4 * 9 = 15 * 144x^4 = 2160x^4 Term 4 (k=3): C(6,3)*(2x)^3*(-3)^3 = 20 * 8x^3 * (-27) = 20 * (-216x^3) = -4320x^3 Term 5 (k=4): C(6,4)*(2x)^2*(-3)^4 = 15 * 4x^2 * 81 = 15 * 324x^2 = 4860x^2 Term 6 (k=5): C(6,5)*(2x)^1*(-3)^5 = 6 * 2x * (-243) = 6 * (-486x) = -2916x Term 7 (k=6): C(6,6)*(2x)^0*(-3)^6 = 1 * 1 * 729 = 729 Step 4: Combine all terms: 64x^6 - 576x^5 + 2160x^4 - 4320x^3 + 4860x^2 - 2916x + 729 The answer is 64x^6 - 576x^5 + 2160x^4 - 4320x^3 + 4860x^2 - 2916x + 729.

6. Expand (5x - 3y)^5 = ? Answer: 3125x^5 - 9375x^4y + 11250x^3y^2 - 6750x^2y^3 + 2025xy^4 - 243y^5 Solution: Use the binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Here a = 5x, b = -3y, n = 5 The binomial coefficients for n=5 are: 1, 5, 10, 10, 5, 1 k=0: C(5,0) * (5x)^5 * (-3y)^0 = 1 * 3125x^5 * 1 = 3125x^5 k=1: C(5,1) * (5x)^4 * (-3y)^1 = 5 * 625x^4 * (-3y) = -9375x^4y k=2: C(5,2) * (5x)^3 *…
   Full step-by-step solution

   Step 1: Use the binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Step 2: Here a = 5x, b = -3y, n = 5 Step 3: The binomial coefficients for n=5 are: 1, 5, 10, 10, 5, 1 Step 4: Expand each term: k=0: C(5,0) * (5x)^5 * (-3y)^0 = 1 * 3125x^5 * 1 = 3125x^5 k=1: C(5,1) * (5x)^4 * (-3y)^1 = 5 * 625x^4 * (-3y) = -9375x^4y k=2: C(5,2) * (5x)^3 * (-3y)^2 = 10 * 125x^3 * 9y^2 = 11250x^3y^2 k=3: C(5,3) * (5x)^2 * (-3y)^3 = 10 * 25x^2 * (-27y^3) = -6750x^2y^3 k=4: C(5,4) * (5x)^1 * (-3y)^4 = 5 * 5x * 81y^4 = 2025xy^4 k=5: C(5,5) * (5x)^0 * (-3y)^5 = 1 * 1 * (-243y^5) = -243y^5 Step 5: Combine all terms: 3125x^5 - 9375x^4y + 11250x^3y^2 - 6750x^2y^3 + 2025xy^4 - 243y^5

7. Olivia is a biomedical researcher studying a newly discovered bacteria. She observes that under a specific culture condition, the bacteria population grows according to the expression (3x + 1)^5, where x represents the number of hours after the initial measurement. To better understand the growth pattern, Olivia needs to expand this expression fully. What is the expanded form of (3x + 1)^5? Answer: 243x^5 + 405x^4 + 270x^3 + 90x^2 + 15x + 1 Solution: Recall the binomial theorem: (a+b)^n = sum_{k=0}^{n} C(n,k) a^(n-k) b^k. Identify a = 3x, b = 1, n = 5. Use Pascal's triangle for n=5: row is 1, 5, 10, 10, 5, 1.
   Full step-by-step solution

   Step 1: Recall the binomial theorem: (a+b)^n = sum_{k=0}^{n} C(n,k) a^(n-k) b^k. Step 2: Identify a = 3x, b = 1, n = 5. Step 3: Use Pascal's triangle for n=5: row is 1, 5, 10, 10, 5, 1. Step 4: Write each term: - k=0: C(5,0)(3x)^5(1)^0 = 1 * (3^5 x^5) * 1 = 243x^5 - k=1: C(5,1)(3x)^4(1)^1 = 5 * (3^4 x^4) * 1 = 5 * 81x^4 = 405x^4 - k=2: C(5,2)(3x)^3(1)^2 = 10 * (3^3 x^3) * 1 = 10 * 27x^3 = 270x^3 - k=3: C(5,3)(3x)^2(1)^3 = 10 * (3^2 x^2) * 1 = 10 * 9x^2 = 90x^2 - k=4: C(5,4)(3x)^1(1)^4 = 5 * (3x) * 1 = 15x - k=5: C(5,5)(3x)^0(1)^5 = 1 * 1 * 1 = 1 Step 5: Combine terms: 243x^5 + 405x^4 + 270x^3 + 90x^2 + 15x + 1 The answer is 243x^5 + 405x^4 + 270x^3 + 90x^2 + 15x + 1.

8. Emma is designing a new acoustic panel for a concert hall. The panel is shaped like a square with side length (3x + 1) meters. To calculate the surface area for material estimation, she needs to expand the expression (3x + 1)^5 using the binomial theorem. What is the expanded form of (3x + 1)^5? Answer: 243x^5 + 405x^4 + 270x^3 + 90x^2 + 15x + 1 Solution: Use the binomial theorem: (a + b)^5 = sum_{k=0}^{5} C(5,k) * a^{5-k} * b^k, where a = 3x and b = 1. Find the binomial coefficients for n = 5 from Pascal's triangle: 1, 5, 10, 10, 5, 1.
   Full step-by-step solution

   Step 1: Use the binomial theorem: (a + b)^5 = sum_{k=0}^{5} C(5,k) * a^{5-k} * b^k, where a = 3x and b = 1. Step 2: Find the binomial coefficients for n = 5 from Pascal's triangle: 1, 5, 10, 10, 5, 1. Step 3: Expand term by term: - k = 0: C(5,0) * (3x)^5 * (1)^0 = 1 * 243x^5 * 1 = 243x^5 - k = 1: C(5,1) * (3x)^4 * (1)^1 = 5 * 81x^4 * 1 = 405x^4 - k = 2: C(5,2) * (3x)^3 * (1)^2 = 10 * 27x^3 * 1 = 270x^3 - k = 3: C(5,3) * (3x)^2 * (1)^3 = 10 * 9x^2 * 1 = 90x^2 - k = 4: C(5,4) * (3x)^1 * (1)^4 = 5 * 3x * 1 = 15x - k = 5: C(5,5) * (3x)^0 * (1)^5 = 1 * 1 * 1 = 1 Step 4: Combine all terms: 243x^5 + 405x^4 + 270x^3 + 90x^2 + 15x + 1 The answer is 243x^5 + 405x^4 + 270x^3 + 90x^2 + 15x + 1.