# Binomial Expansion Calculator

The calculator will find the binomial expansion of the given expression, with steps shown.

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Expand $$\left(2 x + 5\right)^{3}$$$. ## Solution The expansion is given by the following formula: $$\left(a + b\right)^{n} = \sum_{k=0}^{n} {\binom{n}{k}} a^{n - k} b^{k}$$$, where $${\binom{n}{k}} = \frac{n!}{\left(n - k\right)! k!}$$$and $$n! = 1 \cdot 2 \cdot \ldots \cdot n$$$.
We have that $$a = 2 x$$$, $$b = 5$$$, $$n = 3$$$. Therefore, $$\left(2 x + 5\right)^{3} = \sum_{k=0}^{3} {\binom{3}{k}} \left(2 x\right)^{3 - k} 5^{k}$$$.
Now, calculate the product for every value of $$k$$$from $$0$$$ to $$3$$$. $$k = 0$$$: $${\binom{3}{0}} \left(2 x\right)^{3 - 0} \cdot 5^{0} = \frac{3!}{\left(3 - 0\right)! 0!} \left(2 x\right)^{3 - 0} \cdot 5^{0} = 8 x^{3}$$$$$k = 1$$$: $${\binom{3}{1}} \left(2 x\right)^{3 - 1} \cdot 5^{1} = \frac{3!}{\left(3 - 1\right)! 1!} \left(2 x\right)^{3 - 1} \cdot 5^{1} = 60 x^{2}$$$$$k = 2$$$: $${\binom{3}{2}} \left(2 x\right)^{3 - 2} \cdot 5^{2} = \frac{3!}{\left(3 - 2\right)! 2!} \left(2 x\right)^{3 - 2} \cdot 5^{2} = 150 x$$$$$k = 3$$$: $${\binom{3}{3}} \left(2 x\right)^{3 - 3} \cdot 5^{3} = \frac{3!}{\left(3 - 3\right)! 3!} \left(2 x\right)^{3 - 3} \cdot 5^{3} = 125$$$Thus, $$\left(2 x + 5\right)^{3} = 8 x^{3} + 60 x^{2} + 150 x + 125$$$.
$$\left(2 x + 5\right)^{3} = 8 x^{3} + 60 x^{2} + 150 x + 125$$\$A