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Question

Use the Binomial Theorem to expand the binomial.

(d−4b)3

Answer 1

To expand the binomial (d - 4b)^3 using the Binomial Theorem, we use the formula:

(a + b)^n = C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + ... + C(n,n-1)ab^(n-1) + C(n,n)b^n

For (d - 4b)^3, where a = d and b = -4b, we have:

C(3,0)(d^3) + C(3,1)(d^2)(-4b) + C(3,2)(d)(-4b)^2 + C(3,3)(-4b)^3

Now let's calculate the combinations and simplify the expression:

C(3,0) = 1
C(3,1) = 3
C(3,2) = 3
C(3,3) = 1

Substitute these values into the formula:

1(d^3) + 3(d^2)(-4b) + 3(d)(16b^2) + 1(-64b^3)

Simplify:

d^3 - 12d^2b + 48db^2 - 64b^3

Therefore, the expansion of (d - 4b)^3 using the Binomial Theorem is d^3 - 12d^2b + 48db^2 - 64b^3.