Question

Use the Binomial Theorem to expand the binomial
(d + 5b)3

Answer 1

Answer:

$\displaystyle{(d+5b)^3 = d^3 + 15bd^2 + 75b^2d + 125b^3}$

Step-by-step explanation:

The Binomial Theorem states that:

$\displaystyle{(x+y)^n = \binom{n}{0}x^ny^0 + \binom{n}{1}x^{n-1}y^1+\binom{n}{2}x^{n-2}y^2 + \dots + \binom{n}{n}x^{n-n}y^n}$

Note that:

$\displaystyle{_n C _r = \binom{n}{r} = \dfrac{_n P _r}{r!} = \dfrac{n!}{(n-r)!r!}}$

Therefore, first, we will write the expansion:

$\displaystyle{(d+5b)^3 = \binom{3}{0}d^3(5b)^0 + \binom{3}{1}d^2(5b)^1+\binom{3}{2}d^1(5b)^2 + \binom{3}{3}d^0(5b)^3}$

Evaluate each terms:

$\displaystyle{(d+5b)^3=\dfrac{3!}{3!0!}d^3 + \dfrac{3!}{2!1!}d^25b+\dfrac{3!}{1!2!}25b^2d + \dfrac{3!}{0!3!}125b^3}\\\\\displaystyle{(d+5b)^3 = d^3 + 15bd^2 + 75b^2d + 125b^3}$

Henceforth, $\displaystyle{(d+5b)^3 = d^3 + 15bd^2 + 75b^2d + 125b^3}$ is the expansion.