Binomial Theorem

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Answer Key & Explanations

Binomial Theorem · Grade 12 · Worksheet 3

1. Expand (2x - 7y)^4 = ? Answer: 16x^4 - 224x^3y + 1176x^2y^2 - 2744xy^3 + 2401y^4 Solution: Use the binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Here a = 2x, b = -7y, n = 4 The binomial coefficients for n=4 are: 1, 4, 6, 4, 1 k=0: C(4,0) * (2x)^4 * (-7y)^0 = 1 * 16x^4 * 1 = 16x^4 k=1: C(4,1) * (2x)^3 * (-7y)^1 = 4 * 8x^3 * (-7y) = -224x^3y k=2: C(4,2) * (2x)^2 * (-7y)^2 = 6 *…
   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 = 2x, b = -7y, 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) * (2x)^4 * (-7y)^0 = 1 * 16x^4 * 1 = 16x^4 k=1: C(4,1) * (2x)^3 * (-7y)^1 = 4 * 8x^3 * (-7y) = -224x^3y k=2: C(4,2) * (2x)^2 * (-7y)^2 = 6 * 4x^2 * 49y^2 = 1176x^2y^2 k=3: C(4,3) * (2x)^1 * (-7y)^3 = 4 * 2x * (-343y^3) = -2744xy^3 k=4: C(4,4) * (2x)^0 * (-7y)^4 = 1 * 1 * 2401y^4 = 2401y^4 Step 5: Combine all terms: 16x^4 - 224x^3y + 1176x^2y^2 - 2744xy^3 + 2401y^4 The answer is 16x^4 - 224x^3y + 1176x^2y^2 - 2744xy^3 + 2401y^4.

2. Expand (5x - 4y)^6 = ? Answer: 15625x^6 - 75000x^5y + 150000x^4y^2 - 160000x^3y^3 + 96000x^2y^4 - 30720xy^5 + 4096y^6 Solution: Use the binomial theorem: (a + b)^n = Σ(k=0 to n) C(n,k) * a^(n-k) * b^k Here a = 5x, b = -4y, n = 6 The binomial coefficients for n=6 are: 1, 6, 15, 20, 15, 6, 1 k=0: C(6,0) * (5x)^6 * (-4y)^0 = 1 * 15625x^6 * 1 = 15625x^6 k=1: C(6,1) * (5x)^5 * (-4y)^1 = 6 * 3125x^5 * (-4y) = -75000x^5y k=2:…
   Full step-by-step solution

   Step 1: Use the binomial theorem: (a + b)^n = Σ(k=0 to n) C(n,k) * a^(n-k) * b^k Step 2: Here a = 5x, b = -4y, 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) * (5x)^6 * (-4y)^0 = 1 * 15625x^6 * 1 = 15625x^6 k=1: C(6,1) * (5x)^5 * (-4y)^1 = 6 * 3125x^5 * (-4y) = -75000x^5y k=2: C(6,2) * (5x)^4 * (-4y)^2 = 15 * 625x^4 * 16y^2 = 150000x^4y^2 k=3: C(6,3) * (5x)^3 * (-4y)^3 = 20 * 125x^3 * (-64y^3) = -160000x^3y^3 k=4: C(6,4) * (5x)^2 * (-4y)^4 = 15 * 25x^2 * 256y^4 = 96000x^2y^4 k=5: C(6,5) * (5x)^1 * (-4y)^5 = 6 * 5x * (-1024y^5) = -30720xy^5 k=6: C(6,6) * (5x)^0 * (-4y)^6 = 1 * 1 * 4096y^6 = 4096y^6 Step 5: Combine all terms: 15625x^6 - 75000x^5y + 150000x^4y^2 - 160000x^3y^3 + 96000x^2y^4 - 30720xy^5 + 4096y^6

3. Emma is an architect designing a new office building. The building's roof is shaped like a parabola and can be modeled by the function f(x) = (x - 5)^10. She needs to expand this expression to analyze the curvature coefficients for structural support. Expand (x - 5)^10 using the binomial theorem and express your answer as a simplified polynomial in descending powers of x. Answer: x^10 - 50x^9 + 1125x^8 - 15000x^7 + 131250x^6 - 787500x^5 + 3281250x^4 - 9375000x^3 + 17578125x^2 - 19531250x + 9765625 Solution: Identify the binomial form. We have (x - 5)^10, which can be written as (x + (-5))^10. Here, a = x, b = -5, and n = 10.
   Full step-by-step solution

   Step 1: Identify the binomial form. We have (x - 5)^10, which can be written as (x + (-5))^10. Here, a = x, b = -5, and n = 10. Step 2: The binomial theorem states (a+b)^n = sum from k=0 to n of C(n,k) * a^(n-k) * b^k. Step 3: Calculate each term using combinations C(10,k) and powers. C(10,0) * x^10 * (-5)^0 = 1 * x^10 * 1 = x^10 C(10,1) * x^9 * (-5)^1 = 10 * x^9 * (-5) = -50x^9 C(10,2) * x^8 * (-5)^2 = 45 * x^8 * 25 = 1125x^8 C(10,3) * x^7 * (-5)^3 = 120 * x^7 * (-125) = -15000x^7 C(10,4) * x^6 * (-5)^4 = 210 * x^6 * 625 = 131250x^6 C(10,5) * x^5 * (-5)^5 = 252 * x^5 * (-3125) = -787500x^5 C(10,6) * x^4 * (-5)^6 = 210 * x^4 * 15625 = 3281250x^4 C(10,7) * x^3 * (-5)^7 = 120 * x^3 * (-78125) = -9375000x^3 C(10,8) * x^2 * (-5)^8 = 45 * x^2 * 390625 = 17578125x^2 C(10,9) * x^1 * (-5)^9 = 10 * x * (-1953125) = -19531250x C(10,10) * x^0 * (-5)^10 = 1 * 1 * 9765625 = 9765625 Step 4: Combine all terms in descending powers of x: x^10 - 50x^9 + 1125x^8 - 15000x^7 + 131250x^6 - 787500x^5 + 3281250x^4 - 9375000x^3 + 17578125x^2 - 19531250x + 9765625 The answer is the polynomial above.

4. 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.

5. Noah is designing a suspension bridge and needs to model the parabolic shape of the main cable. The cable's height above the deck, in meters, is given by the binomial expansion of (x + 1)^6, where x represents the horizontal distance from the left tower in units of 10 meters. Write the expanded polynomial that models the cable's height, and determine the coefficient of the x^4 term. Answer: 15 Solution: Use the binomial theorem: (x + 1)^6 = sum_{k=0}^{6} C(6, k) * x^(6-k) * 1^k. The term with x^4 occurs when 6 - k = 4, so k = 2. The coefficient is C(6, 2) = 6!
   Full step-by-step solution

   Step 1: Use the binomial theorem: (x + 1)^6 = sum_{k=0}^{6} C(6, k) * x^(6-k) * 1^k. Step 2: The term with x^4 occurs when 6 - k = 4, so k = 2. Step 3: The coefficient is C(6, 2) = 6! / (2! * 4!) = (6 * 5) / (2 * 1) = 30 / 2 = 15. Step 4: The full expansion is: x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1. Step 5: Therefore, the coefficient of the x^4 term is 15. The answer is 15.

6. Emma is constructing a visual pattern using square tiles. The pattern is built by expanding (x + 3y)^5 using the binomial theorem, where x and y represent the side lengths of two different types of square tiles. The expansion produces terms that correspond to the areas of rectangular regions in the visual arrangement. What is the sum of all the coefficients in this expansion? Answer: 1024 Solution: The binomial expansion of (x + 3y)^5 is given by the binomial theorem. The sum of all coefficients can be found by substituting x = 1 and y = 1 into the expression.
   Full step-by-step solution

   Step 1: The binomial expansion of (x + 3y)^5 is given by the binomial theorem. The sum of all coefficients can be found by substituting x = 1 and y = 1 into the expression. Step 2: Substitute x = 1 and y = 1 into (x + 3y)^5: (1 + 3(1))^5 = (1 + 3)^5. Step 3: Simplify inside the parentheses: 1 + 3 = 4. Step 4: Raise to the 5th power: 4^5 = 4 * 4 * 4 * 4 * 4. Step 5: Calculate step by step: 4 * 4 = 16, 16 * 4 = 64, 64 * 4 = 256, 256 * 4 = 1024. Step 6: Therefore, the sum of all coefficients in the expansion is 1024. The answer is 1024.

7. Expand (x + 2y)^5 using the binomial theorem = ? Answer: x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5 Solution: Recall the binomial theorem: (a + b)^n = Σ[k=0 to n] C(n,k) * a^(n-k) * b^k For (x + 2y)^5, we have a = x, b = 2y, n = 5 The binomial coefficients for n=5 are: C(5,0)=1, C(5,1)=5, C(5,2)=10, C(5,3)=10, C(5,4)=5, C(5,5)=1 Term 1: C(5,0) * x^5 * (2y)^0 = 1 * x^5 * 1 = x^5 Term 2: C(5,1) * x^4 *…
   Full step-by-step solution

   Step 1: Recall the binomial theorem: (a + b)^n = Σ[k=0 to n] C(n,k) * a^(n-k) * b^k Step 2: For (x + 2y)^5, we have a = x, b = 2y, n = 5 Step 3: The binomial coefficients for n=5 are: C(5,0)=1, C(5,1)=5, C(5,2)=10, C(5,3)=10, C(5,4)=5, C(5,5)=1 Step 4: Expand term by term: Term 1: C(5,0) * x^5 * (2y)^0 = 1 * x^5 * 1 = x^5 Term 2: C(5,1) * x^4 * (2y)^1 = 5 * x^4 * 2y = 10x^4y Term 3: C(5,2) * x^3 * (2y)^2 = 10 * x^3 * 4y^2 = 40x^3y^2 Term 4: C(5,3) * x^2 * (2y)^3 = 10 * x^2 * 8y^3 = 80x^2y^3 Term 5: C(5,4) * x^1 * (2y)^4 = 5 * x * 16y^4 = 80xy^4 Term 6: C(5,5) * x^0 * (2y)^5 = 1 * 1 * 32y^5 = 32y^5 Step 5: Combine all terms: x^5 + 10x^4y + 40x^3y^2 + 80x^2y^3 + 80xy^4 + 32y^5

8. Expand (4x - 5)^8 = ? Answer: 65536x^8 - 655360x^7 + 2867200x^6 - 7168000x^5 + 11200000x^4 - 11200000x^3 + 7000000x^2 - 2500000x + 390625 Solution: Use the binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Here a = 4x, b = -5, n = 8 The binomial coefficients for n=8 are: 1, 8, 28, 56, 70, 56, 28, 8, 1 k=0: C(8,0) * (4x)^8 * (-5)^0 = 1 * 65536x^8 * 1 = 65536x^8 k=1: C(8,1) * (4x)^7 * (-5)^1 = 8 * 16384x^7 * (-5) = -655360x^7 k=2: C(8,2)…
   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 = -5, n = 8 Step 3: The binomial coefficients for n=8 are: 1, 8, 28, 56, 70, 56, 28, 8, 1 Step 4: Expand each term: k=0: C(8,0) * (4x)^8 * (-5)^0 = 1 * 65536x^8 * 1 = 65536x^8 k=1: C(8,1) * (4x)^7 * (-5)^1 = 8 * 16384x^7 * (-5) = -655360x^7 k=2: C(8,2) * (4x)^6 * (-5)^2 = 28 * 4096x^6 * 25 = 2867200x^6 k=3: C(8,3) * (4x)^5 * (-5)^3 = 56 * 1024x^5 * (-125) = -7168000x^5 k=4: C(8,4) * (4x)^4 * (-5)^4 = 70 * 256x^4 * 625 = 11200000x^4 k=5: C(8,5) * (4x)^3 * (-5)^5 = 56 * 64x^3 * (-3125) = -11200000x^3 k=6: C(8,6) * (4x)^2 * (-5)^6 = 28 * 16x^2 * 15625 = 7000000x^2 k=7: C(8,7) * (4x)^1 * (-5)^7 = 8 * 4x * (-78125) = -2500000x k=8: C(8,8) * (4x)^0 * (-5)^8 = 1 * 1 * 390625 = 390625 Step 5: Combine all terms: 65536x^8 - 655360x^7 + 2867200x^6 - 7168000x^5 + 11200000x^4 - 11200000x^3 + 7000000x^2 - 2500000x + 390625