Question

Use the Binomial Theorem to expand the binomial.

Answer 1

d^3 - 4bd^2 + 16b^2d - 64d^3 is the expanded binomial.

The binomial theorem involves Pascal's triangle, and essentially gives you the coefficients for the formula you're going to use to expand it.

In this case, the coefficients will be 1, 3, 3, and 1.

We can set up our formula like this:

(a + (-b)) = a^3 + a^2b + ab^2 + b^3

Now we can just plug in our values:

(d + (-4b))^3 = d^3 + d^2(-4b) + d(-4b)^2 + (-4b)^3

Now, we can simplify the equation.

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

Answer 2

Answer:

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

Step-by-step explanation:

By the binomial theorem,

$(p+q)^n=\sum_{r=0}^n ^nC_r p^{n-r} q^r$

Where,

$^nC_r=\frac{n!}{(n-r)! r!}$

Thus,

$(d-4b)^3=(d+(-4b))^3=\sum_{r=0}^3 ^3C_r (d)^{3-r} (-4b)^r$

$=^3C_0 (d)^{3-0} (-4b)^0+^3C_1 (d)^{3-1} (-4b)^1+^3C_2 (d)^{3-2} (-4b)^2+^3C_3 (d)^{3-3} (-4b)^3$

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

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