What is expand (2x+y)^5 ?

The solution to expand (2x+y)^5 is 32x^5+80x^4y+80x^3y^2+40x^2y^3+10xy^4+y^5

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expand (2x+y)^5

expand

(2 x + y)^{5}

32 x^{5} + 80 x^{4} y + 80 x^{3} y^{2} + 40 x^{2} y^{3} + 10 x y^{4} + y^{5}

Solution steps

(2 x + y)^{5}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 2 x, b = y

= i = 0 \sum 5 (i 5) (2 x)^{(5 - i)} y^{i}

Expand summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} (2 x)^{5} y^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} (2 x)^{4} y^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} (2 x)^{3} y^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} (2 x)^{2} y^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} (2 x)^{1} y^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} (2 x)^{0} y^{5}

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} (2 x)^{5} y^{0} : 32 x^{5}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} (2 x)^{4} y^{1} : 80 x^{4} y

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (2 x)^{3} y^{2} : 80 x^{3} y^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (2 x)^{2} y^{3} : 40 x^{2} y^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} (2 x)^{1} y^{4} : 10 x y^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} (2 x)^{0} y^{5} : y^{5}

= 32 x^{5} + 80 x^{4} y + 80 x^{3} y^{2} + 40 x^{2} y^{3} + 10 x y^{4} + y^{5}

\frac{- 1}{0}

(2 x - 1) (2 x - 1)

3 (x - 2)^{2}

x^{2^{2}}

4 x - x

What is expand (2x+y)^5 ?
The solution to expand (2x+y)^5 is 32x^5+80x^4y+80x^3y^2+40x^2y^3+10xy^4+y^5