Expand Using the Binomial Theorem (x+5)^8

${(x + 5)}^{8}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{8} \frac{8!}{(8 - k)! k!} \cdot {(x)}^{8 - k} \cdot {(5)}^{k}$

Step 2

Expand the summation.

$\frac{8!}{(8 - 0)! 0!} {(x)}^{8 - 0} \cdot {(5)}^{0} + \frac{8!}{(8 - 1)! 1!} {(x)}^{8 - 1} \cdot {(5)}^{1} + \dots + \frac{8!}{(8 - 8)! 8!} {(x)}^{8 - 8} \cdot {(5)}^{8}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x)}^{8} \cdot {(5)}^{0} + 8 \cdot {(x)}^{7} \cdot {(5)}^{1} + \dots + 1 \cdot {(x)}^{0} \cdot {(5)}^{8}$

Step 4

Simplify the polynomial result.

$x^{8} + 40 x^{7} + 700 x^{6} + 7000 x^{5} + 43750 x^{4} + 175000 x^{3} + 437500 x^{2} + 625000 x + 390625$

Expand Using the Binomial Theorem (x+5)^8

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