How do you use the binomial theorem to expand and simplify the expression (a+5)^5?

Oct 17, 2017

Explanation:

The binomial theorem states that the expansion of ${\left(a + b\right)}^{n} = \left(\begin{matrix}n \\ 0\end{matrix}\right) {a}^{n} {b}^{0} + \left(\begin{matrix}n \\ 1\end{matrix}\right) {a}^{n - 1} b + \ldots + \left(\begin{matrix}n \\ n\end{matrix}\right) {a}^{0} {b}^{n} , n \in \mathbb{N}$

${\left(a + 5\right)}^{5} = \left(\begin{matrix}5 \\ 0\end{matrix}\right) {a}^{5} + \left(\begin{matrix}5 \\ 1\end{matrix}\right) {a}^{4} \cdot 5 + \left(\begin{matrix}5 \\ 2\end{matrix}\right) {a}^{3} \cdot {5}^{2} + \left(\begin{matrix}5 \\ 3\end{matrix}\right) {a}^{2} \cdot {5}^{3} + \left(\begin{matrix}5 \\ 4\end{matrix}\right) {a}^{1} \cdot {5}^{4} + \left(\begin{matrix}5 \\ 5\end{matrix}\right) {a}^{0} \cdot {5}^{5}$
$= {a}^{5} + 25 {a}^{4} + 250 {a}^{3} + 1250 {a}^{2} + 3125 a + 3125$

Oct 17, 2017

${\left(a + 5\right)}^{5} = {a}^{5} + 25 {a}^{4} + 250 {a}^{3} + 1250 {a}^{2} + 3125 a + 3125$

Explanation:

${\left(a + 5\right)}^{5}$
As per Binomial theorem,
${\left(a + b\right)}^{n} = {a}^{n} + n \cdot {a}^{n - 1} \cdot b + \left(\frac{n \left(n - 1\right)}{2}\right) \cdot {a}^{n} - 2 \cdot b 2 + \ldots . + {b}^{n}$
${\left(a + b\right)}^{n} = n C n \cdot {a}^{n} + n C n - 1 \cdot {a}^{n - 1} \cdot b + n C n - 2 \cdot {a}^{n - 2} \cdot {b}^{2} + \ldots . . + {b}^{n}$

Replacing b by 5 and n by 5,
${\left(a + 5\right)}^{5} = a 5 + 5 C 4 \cdot {a}^{4} \cdot 5 + 5 C 3 \cdot {a}^{3} \cdot {5}^{2} + 5 C 2 \cdot {a}^{2} \cdot {5}^{3} + 5 C 1 \cdot a \cdot {5}^{4} + {5}^{5}$

$5 C 1 = 5 C 4 = 5 , 5 C 2 = 5 C 3 = \frac{5 \cdot 4}{1 \cdot 2} = 10$

$= {a}^{5} + 25 {a}^{4} + 250 {a}^{3} + 1250 {a}^{2} + 3125 a + 3125$