Binomial Theorem

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Answer Key & Explanations

Binomial Theorem · Grade 12 · Worksheet 2

1. 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

2. A geometric pattern is formed by expanding (2x - y)^5 using the binomial theorem. The expansion creates terms with coefficients that follow a visual pattern when arranged in triangular form similar to Pascal's triangle. What is the sum of all coefficients in the expansion? Answer: 1 Solution: The binomial expansion of (2x - y)^5 has coefficients that sum to a specific value To find the sum of all coefficients, substitute x = 1 and y = 1 into the expansion This gives us (2(1) - 1)^5 = (2 - 1)^5 = 1^5 1^5 = 1 Therefore, the sum of all coefficients is 1 The answer is 1.
   Full step-by-step solution

   Step 1: The binomial expansion of (2x - y)^5 has coefficients that sum to a specific value Step 2: To find the sum of all coefficients, substitute x = 1 and y = 1 into the expansion Step 3: This gives us (2(1) - 1)^5 = (2 - 1)^5 = 1^5 Step 4: 1^5 = 1 Step 5: Therefore, the sum of all coefficients is 1 The answer is 1.

3. Expand (2x - 7y)^5 = ? Answer: 32x^5 - 560x^4y + 3920x^3y^2 - 13720x^2y^3 + 24010xy^4 - 16807y^5 Solution: Use the binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Here a = 2x, b = -7y, n = 5 The binomial coefficients for n=5 are: 1, 5, 10, 10, 5, 1 k=0: C(5,0) * (2x)^5 * (-7y)^0 = 1 * 32x^5 * 1 = 32x^5 k=1: C(5,1) * (2x)^4 * (-7y)^1 = 5 * 16x^4 * (-7y) = -560x^4y k=2: C(5,2) * (2x)^3 * (-7y)^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 = 2x, b = -7y, 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) * (2x)^5 * (-7y)^0 = 1 * 32x^5 * 1 = 32x^5 k=1: C(5,1) * (2x)^4 * (-7y)^1 = 5 * 16x^4 * (-7y) = -560x^4y k=2: C(5,2) * (2x)^3 * (-7y)^2 = 10 * 8x^3 * 49y^2 = 3920x^3y^2 k=3: C(5,3) * (2x)^2 * (-7y)^3 = 10 * 4x^2 * (-343y^3) = -13720x^2y^3 k=4: C(5,4) * (2x)^1 * (-7y)^4 = 5 * 2x * 2401y^4 = 24010xy^4 k=5: C(5,5) * (2x)^0 * (-7y)^5 = 1 * 1 * (-16807y^5) = -16807y^5 Step 5: Combine all terms: 32x^5 - 560x^4y + 3920x^3y^2 - 13720x^2y^3 + 24010xy^4 - 16807y^5

4. A pharmaceutical company is testing a new drug and knows from clinical trials that the probability of a patient experiencing a specific side effect is 0.15. If a doctor prescribes this medication to 8 patients, what is the probability that exactly 3 of them will experience this side effect? Round your answer to four decimal places. Answer: 0.0839 Solution: We are dealing with a binomial probability problem. P(X = k) = C(n, k) * p^k * (1 - p)^(n - k) - n = number of trials (patients) = 8 - k = number of successes (patients with side effect) = 3 - p = probability of success = 0.15 - C(n, k) = number of combinations of n items taken k at a time = n!
   Full step-by-step solution

   We are dealing with a binomial probability problem. The binomial probability formula is: P(X = k) = C(n, k) * p^k * (1 - p)^(n - k) Where: - n = number of trials (patients) = 8 - k = number of successes (patients with side effect) = 3 - p = probability of success = 0.15 - C(n, k) = number of combinations of n items taken k at a time = n! / (k! * (n - k)!) --- **Step 1: Calculate C(8, 3)** C(8, 3) = 8! / (3! * 5!) = (8 * 7 * 6) / (3 * 2 * 1) = 336 / 6 = 56 --- **Step 2: Calculate p^k** p^k = (0.15)^3 = 0.15 * 0.15 * 0.15 = 0.003375 --- **Step 3: Calculate (1 - p)^(n - k)** 1 - p = 1 - 0.15 = 0.85 n - k = 8 - 3 = 5 (0.85)^5 = 0.85 * 0.85 * 0.85 * 0.85 * 0.85 First: 0.85^2 = 0.7225 Then: 0.7225 * 0.85 = 0.614125 Then: 0.614125 * 0.85 = 0.52200625 Then: 0.52200625 * 0.85 = 0.4437053125 So (0.85)^5 ≈ 0.4437053125 --- **Step 4: Multiply all three parts** P(X = 3) = 56 * 0.003375 * 0.4437053125 First: 56 * 0.003375 = 0.189 Then: 0.189 * 0.4437053125 0.189 * 0.4437053125 = 0.083860303125 --- **Step 5: Round to four decimal places** 0.083860303125 → 0.0839 --- **Final Answer:** 0.0839

5. Emma is designing a solar panel array for a new eco-friendly building. The power output, in kilowatts, of the array is modeled by the expression (2x + 5)^5, where x represents the number of hours of peak sunlight per day. Use the binomial theorem to fully expand this expression, writing it in simplified polynomial form. Answer: 32x^5 + 400x^4 + 2000x^3 + 5000x^2 + 6250x + 3125 Solution: The binomial theorem states: (a + b)^n = sum_{k=0}^{n} C(n,k) a^{n-k} b^k, where C(n,k) = n!/(k!(n-k)!). For (2x + 5)^5, we have n = 5, a = 2x, b = 5.
   Full step-by-step solution

   Step 1: The binomial theorem states: (a + b)^n = sum_{k=0}^{n} C(n,k) a^{n-k} b^k, where C(n,k) = n!/(k!(n-k)!). Step 2: For (2x + 5)^5, we have n = 5, a = 2x, b = 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: Write each term: - Term 1 (k=0): C(5,0)(2x)^5(5)^0 = 1 * 32x^5 * 1 = 32x^5 - Term 2 (k=1): C(5,1)(2x)^4(5)^1 = 5 * 16x^4 * 5 = 5 * 80x^4 = 400x^4 - Term 3 (k=2): C(5,2)(2x)^3(5)^2 = 10 * 8x^3 * 25 = 10 * 200x^3 = 2000x^3 - Term 4 (k=3): C(5,3)(2x)^2(5)^3 = 10 * 4x^2 * 125 = 10 * 500x^2 = 5000x^2 - Term 5 (k=4): C(5,4)(2x)^1(5)^4 = 5 * 2x * 625 = 5 * 1250x = 6250x - Term 6 (k=5): C(5,5)(2x)^0(5)^5 = 1 * 1 * 3125 = 3125 Step 5: Combine all terms: 32x^5 + 400x^4 + 2000x^3 + 5000x^2 + 6250x + 3125 The answer is 32x^5 + 400x^4 + 2000x^3 + 5000x^2 + 6250x + 3125.

6. Sophia is a structural engineer designing a support beam for a pedestrian bridge. The deflection of the beam under a certain load is modeled by the expression (2x - 1)^6, where x is the distance in meters from the left support. To ensure the beam meets safety standards, Sophia needs to expand this expression using the binomial theorem. Write the fully expanded form of (2x - 1)^6. Answer: 64x^6 - 192x^5 + 240x^4 - 160x^3 + 60x^2 - 12x + 1 Solution: Use the binomial theorem: (a + b)^n = sum from k=0 to n of C(n,k) * a^(n-k) * b^k, where a = 2x, b = -1, n = 6. Pascal's triangle for n=6 gives coefficients: 1, 6, 15, 20, 15, 6, 1.
   Full step-by-step solution

   Step 1: Use the binomial theorem: (a + b)^n = sum from k=0 to n of C(n,k) * a^(n-k) * b^k, where a = 2x, b = -1, n = 6. Step 2: Pascal's triangle for n=6 gives coefficients: 1, 6, 15, 20, 15, 6, 1. Step 3: Write each term: - k=0: C(6,0)*(2x)^6*(-1)^0 = 1 * 64x^6 * 1 = 64x^6 - k=1: C(6,1)*(2x)^5*(-1)^1 = 6 * 32x^5 * (-1) = -192x^5 - k=2: C(6,2)*(2x)^4*(-1)^2 = 15 * 16x^4 * 1 = 240x^4 - k=3: C(6,3)*(2x)^3*(-1)^3 = 20 * 8x^3 * (-1) = -160x^3 - k=4: C(6,4)*(2x)^2*(-1)^4 = 15 * 4x^2 * 1 = 60x^2 - k=5: C(6,5)*(2x)^1*(-1)^5 = 6 * 2x * (-1) = -12x - k=6: C(6,6)*(2x)^0*(-1)^6 = 1 * 1 * 1 = 1 Step 4: Combine terms: 64x^6 - 192x^5 + 240x^4 - 160x^3 + 60x^2 - 12x + 1 The answer is 64x^6 - 192x^5 + 240x^4 - 160x^3 + 60x^2 - 12x + 1.

7. Sophia, a botanist, is studying the growth of a rare plant species. She discovers that the number of seeds produced by a plant in a season can be modeled by the expression (2x + 3)^7, where x represents a growth factor related to soil nutrients. To analyze the genetic distribution, she needs to find the coefficient of the x^4 term in the expansion of (2x + 3)^7. Using the binomial theorem, what is this coefficient? Answer: 15120 Solution: Use the binomial theorem: (a + b)^n = sum_{k=0}^n C(n,k) * a^(n-k) * b^k. Here a = 2x, b = 3, n = 7. The general term is T(k+1) = C(7,k) * (2x)^(7-k) * 3^k.
   Full step-by-step solution

   Step 1: Use the binomial theorem: (a + b)^n = sum_{k=0}^n C(n,k) * a^(n-k) * b^k. Here a = 2x, b = 3, n = 7. The general term is T(k+1) = C(7,k) * (2x)^(7-k) * 3^k. Step 2: We want the term with x^4. Since (2x)^(7-k) gives x^(7-k), we set 7 - k = 4, so k = 3. Step 3: For k = 3, the term is T(4) = C(7,3) * (2x)^(4) * 3^3. Step 4: Compute C(7,3) = 7!/(3!4!) = (7*6*5)/(3*2*1) = 210/6 = 35. Step 5: Compute (2x)^4 = 2^4 * x^4 = 16 x^4. Step 6: Compute 3^3 = 27. Step 7: Multiply: coefficient = 35 * 16 * 27 = 35 * 432 = 15120. The answer is 15120.