Question

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

Answer 1

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

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} = (_n P _r)/(r!) = (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=(3!)/(3!0!)d^3 + (3!)/(2!1!)d^25b+(3!)/(1!2!)25b^2d + (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.