What is expand (x+2y)^4 ?

The solution to expand (x+2y)^4 is x^4+8x^3y+24x^2y^2+32xy^3+16y^4

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expand (x+2y)^4

expand

(x + 2 y)^{4}

x^{4} + 8 x^{3} y + 24 x^{2} y^{2} + 32 x y^{3} + 16 y^{4}

Solution steps

(x + 2 y)^{4}

Apply binomial theorem:

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

a = x, b = 2 y

= i = 0 \sum 4 (i 4) x^{(4 - i)} (2 y)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} x^{4} (2 y)^{0} : x^{4}

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} x^{3} (2 y)^{1} : 8 x^{3} y

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} x^{2} (2 y)^{2} : 24 x^{2} y^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} x^{1} (2 y)^{3} : 32 x y^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} x^{0} (2 y)^{4} : 16 y^{4}

= x^{4} + 8 x^{3} y + 24 x^{2} y^{2} + 32 x y^{3} + 16 y^{4}

\frac{31}{4}

8 x^{3}

(3 x^{2} - 10 x + 4) + (10 x^{2} - 5 x + 8)

\frac{1}{7}

(x^{2} - y^{2})^{2}

What is expand (x+2y)^4 ?
The solution to expand (x+2y)^4 is x^4+8x^3y+24x^2y^2+32xy^3+16y^4