Can you check my answers?
1.Use the Binomial Theorem to expand and simplify the expression.
(2 - 3x)3
-27x3 + 104x2 - 36x + 8
-27x3 + 54x2 - 36x + 8
-27x3 + 8
-27x3 + 54x2 - 18x + 8
answer:b
2.Find the 5th term in the expansion of the binomial.
(2x - 4)8
-458,752x3
286,720x4
458,752x3
35,840x4
answer: c
3.Find the 3rd term in the expansion of the binomial.
(x - 2y)12
264x10y2
1760x9y3
-264x10y2
-1760x9y3
answer: d
4.Find the coefficient of the term x2y6 in the binomial expansion of the expression (3x - 2y)8.
81,648
16,128
-81,648
-16,128
answer:c
#1 ok
#2 will be the x^4 term
#3 will be the x^10 y^2 term
#4 Nope. It is
8C6 (3x)^2 (-2y)^6
= 28*3^2*2^6 x^2y^6
= 16128 x^2y^6
Let's check your answers step-by-step.
1.Use the Binomial Theorem to expand and simplify the expression (2 - 3x)3.
The Binomial Theorem states that for any binomial expression (a + b)n, the expansion can be found using the formula:
(a + b)n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a * b^(n-1) + C(n, n) * a^0 * b^n
In our case, a = 2 and b = -3x, so the expansion is:
(2 - 3x)3 = C(3, 0) * 2^3 * (-3x)^0 + C(3, 1) * 2^2 * (-3x)^1 + C(3, 2) * 2^1 * (-3x)^2 + C(3, 3) * 2^0 * (-3x)^3
Let's simplify each term:
= 1 * 2^3 * 1 + 3 * 2^2 * (-3x) + 3 * 2^1 * (-3x)^2 + 1 * 2^0 * (-3x)^3
= 8 - 24x + 36x^2 - 27x^3
So, the correct answer is: -27x^3 + 36x^2 - 24x + 8.
None of the provided options match the correct answer, so none of the answers are correct.
2.Find the 5th term in the expansion of the binomial (2x - 4)8.
We can use the same Binomial Theorem formula:
(a + b)n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a * b^(n-1) + C(n, n) * a^0 * b^n
In this case, a = 2x and b = -4, so the expansion is:
(2x - 4)8 = C(8, 0) * (2x)^8 * (-4)^0 + C(8, 1) * (2x)^7 * (-4)^1 + C(8, 2) * (2x)^6 * (-4)^2 + ...
To find the 5th term, we need to find the coefficient of the (2x)^4 * (-4)^1 term. This corresponds to the term in the expansion when k = 4. So we have:
C(8, 4) * (2x)^4 * (-4)^1 = 70 * (2x)^4 * (-4) = 280x^4 * (-4) = -1120x^4.
Therefore, the 5th term in the expansion is -1120x^4.
None of the provided options match the correct answer, so none of the answers are correct.
3.Find the 3rd term in the expansion of the binomial (x - 2y)12.
Using the Binomial Theorem, we have:
(x - 2y)12 = C(12, 0) * x^12 * (-2y)^0 + C(12, 1) * x^11 * (-2y)^1 + C(12, 2) * x^10 * (-2y)^2 + ...
To find the 3rd term, we need to find the coefficient of the x^10 * (-2y)^2 term. This can be found when k = 2. So we have:
C(12, 2) * x^10 * (-2y)^2 = 66 * x^10 * 4y^2 = 264x^10 * y^2.
Therefore, the 3rd term in the expansion is 264x^10 * y^2.
Option B is the correct answer.
4.Find the coefficient of the term x^2y^6 in the binomial expansion of the expression (3x - 2y)8.
Using the Binomial Theorem, we have:
(3x - 2y)8 = C(8, 0) * (3x)^8 * (-2y)^0 + C(8, 1) * (3x)^7 * (-2y)^1 + C(8, 2) * (3x)^6 * (-2y)^2 + ...
To find the term with x^2y^6, we need to find the coefficient when k = 6. So we have:
C(8, 6) * (3x)^2 * (-2y)^6 = 28 * (9x^2) * 64y^6 = 16128x^2 * y^6.
Therefore, the coefficient of the term x^2y^6 is 16128.
Option C is the correct answer.
To check your answers for the binomial expansions, we can use the Binomial Theorem.
1. To expand (2 - 3x)³, we raise the first term, 2, to the third power, then multiply by the first term, second term, and third term raised to the appropriate powers, respectively:
(2 - 3x)³ = 2³ - 3(2²)(3x) + 3(2)(3x)² - (3x)³
Simplifying, we get:
8 - 12x + 36x² - 27x³
The correct answer is: -27x³ + 36x² - 12x + 8. So your answer is incorrect.
2. To find the 5th term in the expansion of (2x - 4)⁸, we can use the formula for the general term in the binomial expansion:
T(n) = (nCr)(a^(n-r))(b^r)
where n is the power of the binomial, r is the term number starting from 0, a is the coefficient of the first term, and b is the coefficient of the second term.
In this case, n = 8, r = 4 (since we're looking for the 5th term), a = 2x, and b = -4.
Plugging in the values, we get:
T(4) = (8C4)(2x)^(8-4)(-4)^4 = 70 * 16 * 256 = 286,720x⁴.
So your answer is correct.
3. To find the 3rd term in the expansion of (x - 2y)¹², we use the same formula:
T(n) = (nCr)(a^(n-r))(b^r)
In this case, n = 12, r = 2 (since we're looking for the 3rd term), a = x, and b = -2y.
Plugging in the values, we get:
T(2) = (12C2)(x)^(12-2)(-2y)^2 = 66 * x¹⁰ * 4y² = 264x¹⁰y².
So your answer is correct.
4. To find the coefficient of the term x²y⁶ in the expansion of (3x - 2y)⁸, we use the same formula:
T(n) = (nCr)(a^(n-r))(b^r)
In this case, n = 8, r = 6, a = 3x, and b = -2y.
Plugging in the values, we get:
T(6) = (8C6)(3x)^(8-6)(-2y)⁶ = 28 * 9x² * 64y⁶ = 16,128x²y⁶.
So your answer is correct.
Overall, you got 2 out of 4 answers correct. Keep practicing and using the Binomial Theorem to improve your understanding!