Binomial Theorem

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Answer Key & Explanations

Binomial Theorem · Grade 12 · Worksheet 1

1. Hana is constructing a visual pattern based on the binomial expansion of (2x + 3y)^7. She arranges the terms in order of descending powers of x, forming a pyramid shape. Each term in the pyramid has a coefficient that can be found using the binomial theorem. What is the coefficient of the x^4 y^3 term in the expansion? Answer: 15120 Solution: The binomial theorem states that (a + b)^n = sum_{k=0}^{n} C(n, k) * a^(n-k) * b^k. Here, a = 2x, b = 3y, and n = 7. The general term is C(7, k) * (2x)^(7-k) * (3y)^k.
   Full step-by-step solution

   Step 1: The binomial theorem states that (a + b)^n = sum_{k=0}^{n} C(n, k) * a^(n-k) * b^k. Step 2: Here, a = 2x, b = 3y, and n = 7. The general term is C(7, k) * (2x)^(7-k) * (3y)^k. Step 3: We need the term where the power of x is 4 and the power of y is 3. Since x comes from a = 2x, its power is 7 - k, so 7 - k = 4 gives k = 3. Step 4: For k = 3, the term is C(7, 3) * (2x)^(7-3) * (3y)^3 = C(7, 3) * (2x)^4 * (3y)^3. Step 5: Calculate C(7, 3) = 7! / (3! * 4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35. Step 6: (2x)^4 = 2^4 * x^4 = 16 x^4. (3y)^3 = 3^3 * y^3 = 27 y^3. Step 7: Multiply the coefficient parts: 35 * 16 * 27 = 35 * 432 = 15120. Step 8: Therefore, the coefficient of the x^4 y^3 term is 15120. The answer is 15120.

2. Olivia is a geneticist studying a population of 1000 plants. She knows that the probability of a single plant having a specific gene mutation is 0.15. If she randomly selects 10 plants for a detailed genetic analysis, what is the probability that exactly 3 of them will have the mutation? Use the binomial probability formula and round your answer to four decimal places. Answer: 0.1298 Solution: Identify the binomial probability formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k). Step 2: Extract values: n = 10 (number of plants), k = 3 (number with mutation), p = 0.15 (probability of mutation), 1-p = 0.85.
   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 values: n = 10 (number of plants), k = 3 (number with mutation), p = 0.15 (probability of mutation), 1-p = 0.85. Step 3: Calculate the combination: C(10, 3) = 10! / (3! * 7!) = (10 * 9 * 8) / (3 * 2 * 1) = 720 / 6 = 120. Step 4: Calculate p^k: (0.15)^3 = 0.003375. Step 5: Calculate (1-p)^(n-k): (0.85)^7. Compute stepwise: 0.85^2 = 0.7225, 0.85^3 = 0.614125, 0.85^4 = 0.52200625, 0.85^5 = 0.4437053125, 0.85^6 = 0.377149515625, 0.85^7 = 0.32057708828125. Step 6: Multiply all terms: 120 * 0.003375 * 0.32057708828125 = 0.405 * 0.32057708828125 = 0.12983372075390625. Step 7: Round to four decimal places: 0.1298. The answer is 0.1298.

3. Mason is designing a complex electrical circuit for a physics project. The current flowing through a specific branch of the circuit, measured in amperes, is given by the expression (2x - 5)^9, where x represents a variable resistance in ohms. Mason needs to expand this expression to analyze the circuit's behavior. Use the binomial theorem to fully expand (2x - 5)^9 and write the simplified polynomial in descending powers of x. Answer: 512x^9 - 11520x^8 + 115200x^7 - 672000x^6 + 2520000x^5 - 6300000x^4 + 10500000x^3 - 11250000x^2 + 7031250x - 1953125 Solution: Identify a = 2x, b = -5, n = 9. Use the binomial theorem: (a + b)^n = sum_{k=0}^{n} C(n,k) * a^(n-k) * b^k. Step 2: Find coefficients from Pascal's triangle for n = 9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1.
   Full step-by-step solution

   Step 1: Identify a = 2x, b = -5, n = 9. Use the binomial theorem: (a + b)^n = sum_{k=0}^{n} C(n,k) * a^(n-k) * b^k. Step 2: Find coefficients from Pascal's triangle for n = 9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1. Step 3: Expand each term: Term 0: C(9,0) * (2x)^9 * (-5)^0 = 1 * 512x^9 * 1 = 512x^9 Term 1: C(9,1) * (2x)^8 * (-5)^1 = 9 * 256x^8 * (-5) = -11520x^8 Term 2: C(9,2) * (2x)^7 * (-5)^2 = 36 * 128x^7 * 25 = 115200x^7 Term 3: C(9,3) * (2x)^6 * (-5)^3 = 84 * 64x^6 * (-125) = -672000x^6 Term 4: C(9,4) * (2x)^5 * (-5)^4 = 126 * 32x^5 * 625 = 2520000x^5 Term 5: C(9,5) * (2x)^4 * (-5)^5 = 126 * 16x^4 * (-3125) = -6300000x^4 Term 6: C(9,6) * (2x)^3 * (-5)^6 = 84 * 8x^3 * 15625 = 10500000x^3 Term 7: C(9,7) * (2x)^2 * (-5)^7 = 36 * 4x^2 * (-78125) = -11250000x^2 Term 8: C(9,8) * (2x)^1 * (-5)^8 = 9 * 2x * 390625 = 7031250x Term 9: C(9,9) * (2x)^0 * (-5)^9 = 1 * 1 * (-1953125) = -1953125 Step 4: Combine all terms in descending powers of x: 512x^9 - 11520x^8 + 115200x^7 - 672000x^6 + 2520000x^5 - 6300000x^4 + 10500000x^3 - 11250000x^2 + 7031250x - 1953125 The answer is 512x^9 - 11520x^8 + 115200x^7 - 672000x^6 + 2520000x^5 - 6300000x^4 + 10500000x^3 - 11250000x^2 + 7031250x - 1953125.

4. 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.

5. 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.

6. Aroha is constructing a visual proof of the binomial theorem using square tiles. She arranges the expansion of (3a + 4b)^8 into a rectangular grid where each term's coefficient is represented by a distinct color. The grid is organized such that the terms with the highest power of 'a' are in the top row, and the power of 'a' decreases by 1 each row. If the number of tiles of each color corresponds exactly to the coefficient of that term, what is the total number of square tiles Aroha uses to represent the entire expansion? Answer: 5764801 Solution: The sum of all coefficients in the expansion of (3a + 4b)^8 is found by substituting a = 1 and b = 1 into the expression.
   Full step-by-step solution

   Step 1: The sum of all coefficients in the expansion of (3a + 4b)^8 is found by substituting a = 1 and b = 1 into the expression. This works because each term's coefficient is multiplied by a^m b^n, and when a = b = 1, the powers become 1, leaving only the coefficients. Step 2: Substitute a = 1 and b = 1 into (3a + 4b)^8: (3*1 + 4*1)^8 = (3 + 4)^8. Step 3: Calculate 3 + 4 = 7. Step 4: Raise 7 to the 8th power: 7^8 = 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7. Step 5: Compute step by step: 7^2 = 49, 7^4 = 49 * 49 = 2401, 7^8 = 2401 * 2401 = 5,764,801. Step 6: Therefore, the total number of tiles is 5,764,801. The answer is 5764801.