Whats the right answer
Use the Binomial Theorem to expand the binomial.
(d – 4b)3
The expanded form of (d - 4b)^3 using the Binomial Theorem is:
(d - 4b)^3 = d^3 - 3d^2(4b) + 3d(4b)^2 - (4b)^3
Simplifying this expression further:
= d^3 - 12d^2b + 48db^2 - 64b^3
To expand the binomial (d – 4b) raised to the power of 3 using the Binomial Theorem, follow these steps:
Step 1: Identify the values for n (the exponent) and r (the term number). In this case, n = 3 and r can range from 0 to 3.
Step 2: Use the formula for the Binomial Coefficient, also known as "n choose r," which is given by nCr = n! / (r!(n-r)!), where n! denotes the factorial of n.
Step 3: Write out the expansion using the formula: (d – 4b)3 = (nCr)(d)n-r(-4b)r.
Step 4: Substitute the values of n, r, d, and -4b into the formula.
Expanding (d – 4b)3:
Term 1: (3C0)(d)3-0(-4b)0
= (1)(d^3)(1)
= d^3
Term 2: (3C1)(d)3-1(-4b)1
= (3)(d^2)(-4b)
= -12bd^2
Term 3: (3C2)(d)3-2(-4b)2
= (3)(d^1)(16b^2)
= 48b^2d
Term 4: (3C3)(d)3-3(-4b)3
= (1)(1)(-64b^3)
= -64b^3
Therefore, the expanded form of (d – 4b)3 is:
d^3 - 12bd^2 + 48b^2d - 64b^3
To expand the binomial (d – 4b) raised to the power of 3 using the Binomial Theorem, we can follow these steps:
Step 1: Identify the values of n and k.
In this case, n = 3 (the exponent) and k represents the term number. We will start from k = 0 and go up to k = 3.
Step 2: Write down the binomial coefficients.
The binomial coefficients can be calculated using the combination formula, which is given by: nCk = n! / (k! * (n - k)!). For each value of k, we will calculate nCk.
nC0 = 3! / (0! * (3 - 0)!) = 3! / (0! * 3!) = 1
nC1 = 3! / (1! * (3 - 1)!) = 3! / (1! * 2!) = 3
nC2 = 3! / (2! * (3 - 2)!) = 3! / (2! * 1!) = 3
nC3 = 3! / (3! * (3 - 3)!) = 3! / (3! * 0!) = 1
Step 3: Expand the binomial using the terms.
Now we can write the expanded form using the terms and the binomial coefficients.
(d – 4b)3 = (1)(d)^3 + (3)(d)^2(-4b) + (3)(d)(-4b)^2 + (1)(-4b)^3
Simplifying:
(d – 4b)3 = d^3 - 12d^2b + 48db^2 - 64b^3
Therefore, the expanded form of (d – 4b)3 using the Binomial Theorem is d^3 - 12d^2b + 48db^2 - 64b^3.