Question

Expand the binomial using the Binomial Theorem.
(3c − d)^3

Answer 1

`27c^3-27c^2d+9cd^2-d^3`

Explanation:

Binomial Theorem

`(a+b)^n=a^n+(n!)/(1!(n-1)!)a^(n-1)b+(n!)/(2!(n-2)!)a^(n-2)b^2+...+(n!)/(r!(n-r)!)a^(n-r)b^r+...+b^n`

`\textsf{If }(a+b)^n=(3c-d)^3 \: \textsf{ then}:`

• a = 3c
• b = -d
• n = 3

Substitute the given values into the formula:

`\begin{aligned}(3c-d)^3 & =(3c)^3+(3!)/(1!(3-1)!)(3c)^(3-1)(-d)+(3!)/(2!(3-2)!)(3c)^(3-2)(-d)^2+(-d)^3\\\\& =(3c)^3+3(3c)^2(-d)+3(3c)^1(-d)^2+(-d)^3\\\\& =27c^3-27c^2d+9cd^2-d^3\end{aligned}`