Binomial Theorem

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Answer Key & Explanations

Binomial Theorem · Grade 12 · Worksheet 2

1. Olivia is a financial analyst studying the growth of an investment fund. The value of the fund, in thousands of dollars, is modeled by the expression (3x + 1)^5, where x represents the number of years since the fund was established. Using the binomial theorem, expand (3x + 1)^5 completely to determine the polynomial that represents the fund's value over time. Answer: 243x^5 + 405x^4 + 270x^3 + 90x^2 + 15x + 1 Solution: Identify a = 3x, b = 1, and n = 5. Use the binomial theorem: (a + b)^n = sum_{k=0}^{n} C(n,k) * a^(n-k) * b^k. The coefficients from Pascal's triangle for n = 5 are: 1, 5, 10, 10, 5, 1.
   Full step-by-step solution

   Step 1: Identify a = 3x, b = 1, and n = 5. Use the binomial theorem: (a + b)^n = sum_{k=0}^{n} C(n,k) * a^(n-k) * b^k. Step 2: The coefficients from Pascal's triangle for n = 5 are: 1, 5, 10, 10, 5, 1. Step 3: Write each 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.

2. Expand (4x - 3y)^8 = ? Answer: 65536x^8 - 393216x^7y + 1032192x^6y^2 - 1548288x^5y^3 + 1451520x^4y^4 - 870912x^3y^5 + 326592x^2y^6 - 69984xy^7 + 6561y^8 Solution: Use binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Here a = 4x, b = -3y, n = 8 Binomial coefficients for n=8: 1, 8, 28, 56, 70, 56, 28, 8, 1 - k=0: C(8,0) * (4x)^8 * (-3y)^0 = 1 * 65536x^8 * 1 = 65536x^8 - k=1: C(8,1) * (4x)^7 * (-3y)^1 = 8 * 16384x^7 * (-3y) = -393216x^7y - k=2: C(8,2)…
   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 = -3y, n = 8 Step 3: Binomial coefficients for n=8: 1, 8, 28, 56, 70, 56, 28, 8, 1 Step 4: Expand each term: - k=0: C(8,0) * (4x)^8 * (-3y)^0 = 1 * 65536x^8 * 1 = 65536x^8 - k=1: C(8,1) * (4x)^7 * (-3y)^1 = 8 * 16384x^7 * (-3y) = -393216x^7y - k=2: C(8,2) * (4x)^6 * (-3y)^2 = 28 * 4096x^6 * 9y^2 = 1032192x^6y^2 - k=3: C(8,3) * (4x)^5 * (-3y)^3 = 56 * 1024x^5 * (-27y^3) = -1548288x^5y^3 - k=4: C(8,4) * (4x)^4 * (-3y)^4 = 70 * 256x^4 * 81y^4 = 1451520x^4y^4 - k=5: C(8,5) * (4x)^3 * (-3y)^5 = 56 * 64x^3 * (-243y^5) = -870912x^3y^5 - k=6: C(8,6) * (4x)^2 * (-3y)^6 = 28 * 16x^2 * 729y^6 = 326592x^2y^6 - k=7: C(8,7) * (4x)^1 * (-3y)^7 = 8 * 4x * (-2187y^7) = -69984xy^7 - k=8: C(8,8) * (4x)^0 * (-3y)^8 = 1 * 1 * 6561y^8 = 6561y^8 Step 5: Combine all terms: 65536x^8 - 393216x^7y + 1032192x^6y^2 - 1548288x^5y^3 + 1451520x^4y^4 - 870912x^3y^5 + 326592x^2y^6 - 69984xy^7 + 6561y^8

3. Expand (5x - 3)^4 = ? Answer: 625x^4 - 1500x^3 + 1350x^2 - 540x + 81 Solution: Use binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Here a = 5x, b = -3, n = 4 Binomial coefficients for n=4 are: 1, 4, 6, 4, 1 k=0: C(4,0) * (5x)^4 * (-3)^0 = 1 * 625x^4 * 1 = 625x^4 k=1: C(4,1) * (5x)^3 * (-3)^1 = 4 * 125x^3 * (-3) = -1500x^3 k=2: C(4,2) * (5x)^2 * (-3)^2 = 6 * 25x^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 = 5x, b = -3, n = 4 Step 3: Binomial coefficients for n=4 are: 1, 4, 6, 4, 1 Step 4: Expand each term: k=0: C(4,0) * (5x)^4 * (-3)^0 = 1 * 625x^4 * 1 = 625x^4 k=1: C(4,1) * (5x)^3 * (-3)^1 = 4 * 125x^3 * (-3) = -1500x^3 k=2: C(4,2) * (5x)^2 * (-3)^2 = 6 * 25x^2 * 9 = 1350x^2 k=3: C(4,3) * (5x)^1 * (-3)^3 = 4 * 5x * (-27) = -540x k=4: C(4,4) * (5x)^0 * (-3)^4 = 1 * 1 * 81 = 81 Step 5: Combine all terms: 625x^4 - 1500x^3 + 1350x^2 - 540x + 81

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. A quality control engineer at a microchip manufacturing plant is testing a new production line. The engineer knows from initial testing that the probability of any individual microchip being defective is 0.08. If the engineer randomly selects 12 microchips from the production line for detailed inspection, what is the probability that exactly 2 of them will be defective? Use the binomial probability formula and round your answer to four decimal places. Answer: 0.1835 Solution: Identify the binomial probability formula: P(X = k) = C(n,k) * p^k * (1-p)^(n-k) Extract the values from the problem: n = 12 (number of trials), k = 2 (number of successes), p = 0.08 (probability of success) Calculate the combination: C(12,2) = 12!/(2!*10!) = (12*11)/(2*1) = 132/2 = 66 Calculate…
   Full step-by-step solution

   Step 1: Identify the binomial probability formula: P(X = k) = C(n,k) * p^k * (1-p)^(n-k) Step 2: Extract the values from the problem: n = 12 (number of trials), k = 2 (number of successes), p = 0.08 (probability of success) Step 3: Calculate the combination: C(12,2) = 12!/(2!*10!) = (12*11)/(2*1) = 132/2 = 66 Step 4: Calculate p^k: (0.08)^2 = 0.0064 Step 5: Calculate (1-p)^(n-k): (1-0.08)^(12-2) = (0.92)^10 ≈ 0.434388 Step 6: Multiply all terms: 66 * 0.0064 * 0.434388 ≈ 0.1835 Step 7: Round to four decimal places: 0.1835 The answer is 0.1835.

6. Expand (4x - 5)^5 = ? Answer: 1024x^5 - 6400x^4 + 16000x^3 - 20000x^2 + 12500x - 3125 Solution: Use the binomial theorem: (a + b)^n = Σ C(n,k) * a^(n-k) * b^k Here a = 4x, b = -5, n = 5 The binomial coefficients for n=5 are: 1, 5, 10, 10, 5, 1 k=0: C(5,0) * (4x)^5 * (-5)^0 = 1 * 1024x^5 * 1 = 1024x^5 k=1: C(5,1) * (4x)^4 * (-5)^1 = 5 * 256x^4 * (-5) = -6400x^4 k=2: C(5,2) * (4x)^3 * (-5)^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 = 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) * (4x)^5 * (-5)^0 = 1 * 1024x^5 * 1 = 1024x^5 k=1: C(5,1) * (4x)^4 * (-5)^1 = 5 * 256x^4 * (-5) = -6400x^4 k=2: C(5,2) * (4x)^3 * (-5)^2 = 10 * 64x^3 * 25 = 16000x^3 k=3: C(5,3) * (4x)^2 * (-5)^3 = 10 * 16x^2 * (-125) = -20000x^2 k=4: C(5,4) * (4x)^1 * (-5)^4 = 5 * 4x * 625 = 12500x k=5: C(5,5) * (4x)^0 * (-5)^5 = 1 * 1 * (-3125) = -3125 Step 5: Combine all terms: 1024x^5 - 6400x^4 + 16000x^3 - 20000x^2 + 12500x - 3125

7. Tane is a biomedical engineer designing a new type of stent for a coronary artery. The stent's expansion force, measured in newtons, is modeled by the expression (3x + 1)^5, where x represents the radial expansion distance in millimeters. To ensure the stent provides uniform support, Tane must fully expand this expression 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: Identify a = 3x, b = 1, n = 5. The binomial theorem states (a+b)^5 = sum_{k=0}^5 C(5,k) * a^(5-k) * b^k. The coefficients from Pascal's triangle for n=5 are: 1, 5, 10, 10, 5, 1.
   Full step-by-step solution

   Step 1: Identify a = 3x, b = 1, n = 5. Step 2: The binomial theorem states (a+b)^5 = sum_{k=0}^5 C(5,k) * a^(5-k) * b^k. Step 3: The coefficients from Pascal's triangle for n=5 are: 1, 5, 10, 10, 5, 1. Step 4: Write each 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 5: Add all terms: 243x^5 + 405x^4 + 270x^3 + 90x^2 + 15x + 1 The expanded form is 243x^5 + 405x^4 + 270x^3 + 90x^2 + 15x + 1.

8. Expand (2x - 3)^4 using the binomial theorem = ? Answer: 16x^4 - 96x^3 + 216x^2 - 216x + 81 Solution: We expand (2x - 3)^4 using the binomial theorem. (a + b)^n = sum_{k=0}^{n} [ C(n,k) * a^(n-k) * b^k ] Here: a = 2x, b = -3, n = 4. Write the general term for k = 0 to 4.
   Full step-by-step solution

   We expand (2x - 3)^4 using the binomial theorem. The binomial theorem says: (a + b)^n = sum_{k=0}^{n} [ C(n,k) * a^(n-k) * b^k ] Here: a = 2x, b = -3, n = 4. Step 1: Write the general term for k = 0 to 4. Term k = C(4,k) * (2x)^(4-k) * (-3)^k Step 2: Compute each term for k = 0, 1, 2, 3, 4. k = 0: C(4,0) = 1 (2x)^(4-0) = (2x)^4 = 16 x^4 (-3)^0 = 1 Term = 1 * 16 x^4 * 1 = 16 x^4 k = 1: C(4,1) = 4 (2x)^(4-1) = (2x)^3 = 8 x^3 (-3)^1 = -3 Term = 4 * 8 x^3 * (-3) = 4 * 8 * (-3) x^3 = -96 x^3 k = 2: C(4,2) = 6 (2x)^(4-2) = (2x)^2 = 4 x^2 (-3)^2 = 9 Term = 6 * 4 x^2 * 9 = 6 * 4 * 9 x^2 = 216 x^2 k = 3: C(4,3) = 4 (2x)^(4-3) = (2x)^1 = 2x (-3)^3 = -27 Term = 4 * 2x * (-27) = 4 * 2 * (-27) x = -216 x k = 4: C(4,4) = 1 (2x)^(4-4) = (2x)^0 = 1 (-3)^4 = 81 Term = 1 * 1 * 81 = 81 Step 3: Add all terms: 16 x^4 + (-96 x^3) + 216 x^2 + (-216 x) + 81 Step 4: Write final expanded form: 16 x^4 - 96 x^3 + 216 x^2 - 216 x + 81 This matches the correct answer.