Use the Binomial Theorem to expand the binomial:
(d – 4b)^3
The Binomial Theorem Quick Check:
1. A. or the one that has "-1,458d + 729" at the end of it.
2. B. or the one that has "+243v^5" at the end of it.
3. D. or "-64b^3"
100% 3/3
To expand the binomial (d - 4b)^3 using the Binomial Theorem, we can use 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^1 * b^(n-1) + C(n, n) * a^0 * b^n
In this case, a = d and b = -4b.
The binomial coefficients C(n, k) are calculated using the formula:
C(n, k) = n! / (k! * (n-k)!)
Let's expand (d - 4b)^3 using the Binomial Theorem step-by-step:
Step 1: Determine the value of n in (d - 4b)^3
In this case, n = 3
Step 2: Determine the values of k from 0 to n
In this case, k takes on the values 0, 1, 2, and 3 for n = 3
Step 3: Calculate C(n, k) for each value of k
C(3, 0) = 3! / (0! * (3-0)!) = 1
C(3, 1) = 3! / (1! * (3-1)!) = 3
C(3, 2) = 3! / (2! * (3-2)!) = 3
C(3, 3) = 3! / (3! * (3-3)!) = 1
Step 4: Plug in the values of a, b, and k into the formula
Using the formula:
(d - 4b)^3 = C(3, 0) * d^3 * (-4b)^0 + C(3, 1) * d^2 * (-4b)^1 + C(3, 2) * d^1 * (-4b)^2 + C(3, 3) * d^0 * (-4b)^3
Step 5: Simplify each term
(d - 4b)^3 = 1 * d^3 * 1 + 3 * d^2 * (-4b) + 3 * d^1 * (-4b)^2 + 1 * 1 * (-4b)^3
Step 6: Simplify the terms further
(d - 4b)^3 = d^3 - 12bd^2 + 48b^2d - 64b^3
Therefore, the expanded form of (d - 4b)^3 using the Binomial Theorem is d^3 - 12bd^2 + 48b^2d - 64b^3.
To expand the binomial (d - 4b)^3 using the Binomial Theorem, we can follow these steps:
Step 1: Determine the values of n and k.
Here, n = 3 (because the exponent of the binomial is 3), and k will range from 0 to 3.
Step 2: Write down the Binomial Theorem formula.
The Binomial Theorem states that "(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,k) * a^(n-k) * b^k", where C(n,k) denotes the binomial coefficient, which is calculated as "n! / (k!(n-k)!)".
Step 3: Calculate the binomial coefficients.
For each value of k (ranging from 0 to 3), calculate the binomial coefficient C(3, k) using the formula mentioned earlier. The binomial coefficients are as follows:
C(3, 0) = 1
C(3, 1) = 3
C(3, 2) = 3
C(3, 3) = 1
Step 4: Substitute the values into the formula.
Substitute the calculated binomial coefficients and the variables into the Binomial Theorem formula. In this case, let a = d and b = -4b.
The expanded form of (d - 4b)^3 using the Binomial Theorem is:
1 * d^3 * (-4b)^0 + 3 * d^2 * (-4b)^1 + 3 * d^1 * (-4b)^2 + 1 * d^0 * (-4b)^3
Step 5: Simplify the expression.
Simplify the exponents and resolve any calculations. In this case, (-4b)^0 = 1, (-4b)^1 = -4b, (-4b)^2 = 16b^2, and (-4b)^3 = -64b^3.
The expanded form becomes:
d^3 + 3 * d^2 * (-4b) + 3 * d * 16b^2 + (-64b^3)
Finally, simplify the expression further if needed.