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Popular >

expand (a+b+c+d+1)^3

Solution

expand

(a + b + c + d + 1)^{3}

Solution

a^{3} + 3 a^{2} b + 3 a^{2} c + 3 a^{2} d + 3 a^{2} + 3 a b^{2} + 3 b^{2} c + 3 b^{2} d + 6 ab + 3 a c^{2} + 6 a c + 3 a d^{2} + 6 a d + 6 ab c + 6 ab d + 6 a c d + 3 a + b^{3} + 3 b^{2} + 3 b c^{2} + 6 b c + 3 b d^{2} + 6 b d + 6 b c d + 3 b + c^{3} + 3 c^{2} d + 3 c^{2} + 3 c d^{2} + 6 c d + 3 c + d^{3} + 3 d^{2} + 3 d + 1

Solution steps

(a + b + c + d + 1)^{3}

Simplify

(a + b + c + d + 1)^{3} : (a + b + c + d + 1) (a + b + c + d + 1) (a + b + c + d + 1)

= (a + b + c + d + 1) (a + b + c + d + 1) (a + b + c + d + 1)

Expand

(a + b + c + d + 1) (a + b + c + d + 1) : a^{2} + 2 ab + 2 a c + 2 a d + 2 a + b^{2} + 2 b c + 2 b d + 2 b + c^{2} + 2 c d + 2 c + d^{2} + 2 d + 1

= (a^{2} + 2 ab + 2 a c + 2 a d + 2 a + b^{2} + 2 b c + 2 b d + 2 b + c^{2} + 2 c d + 2 c + d^{2} + 2 d + 1) (a + b + c + d + 1)

Distribute parentheses

= a^{2} a + a^{2} b + a^{2} c + a^{2} d + a^{2} \cdot 1 + 2 aba + 2 abb + 2 ab c + 2 ab d + 2 ab \cdot 1 + 2 a c a + 2 a c b + 2 a cc + 2 a c d + 2 a c \cdot 1 + 2 a d a + 2 a d b + 2 a d c + 2 a dd + 2 a d \cdot 1 + 2 aa + 2 ab + 2 a c + 2 a d + 2 a \cdot 1 + b^{2} a + b^{2} b + b^{2} c + b^{2} d + b^{2} \cdot 1 + 2 b c a + 2 b c b + 2 b cc + 2 b c d + 2 b c \cdot 1 + 2 b d a + 2 b d b + 2 b d c + 2 b dd + 2 b d \cdot 1 + 2 ba + 2 bb + 2 b c + 2 b d + 2 b \cdot 1 + c^{2} a + c^{2} b + c^{2} c + c^{2} d + c^{2} \cdot 1 + 2 c d a + 2 c d b + 2 c d c + 2 c dd + 2 c d \cdot 1 + 2 c a + 2 c b + 2 cc + 2 c d + 2 c \cdot 1 + d^{2} a + d^{2} b + d^{2} c + d^{2} d + d^{2} \cdot 1 + 2 d a + 2 d b + 2 d c + 2 dd + 2 d \cdot 1 + 1 \cdot a + 1 \cdot b + 1 \cdot c + 1 \cdot d + 1 \cdot 1

Simplify

a^{2} a + a^{2} b + a^{2} c + a^{2} d + a^{2} \cdot 1 + 2 aba + 2 abb + 2 ab c + 2 ab d + 2 ab \cdot 1 + 2 a c a + 2 a c b + 2 a cc + 2 a c d + 2 a c \cdot 1 + 2 a d a + 2 a d b + 2 a d c + 2 a dd + 2 a d \cdot 1 + 2 aa + 2 ab + 2 a c + 2 a d + 2 a \cdot 1 + b^{2} a + b^{2} b + b^{2} c + b^{2} d + b^{2} \cdot 1 + 2 b c a + 2 b c b + 2 b cc + 2 b c d + 2 b c \cdot 1 + 2 b d a + 2 b d b + 2 b d c + 2 b dd + 2 b d \cdot 1 + 2 ba + 2 bb + 2 b c + 2 b d + 2 b \cdot 1 + c^{2} a + c^{2} b + c^{2} c + c^{2} d + c^{2} \cdot 1 + 2 c d a + 2 c d b + 2 c d c + 2 c dd + 2 c d \cdot 1 + 2 c a + 2 c b + 2 cc + 2 c d + 2 c \cdot 1 + d^{2} a + d^{2} b + d^{2} c + d^{2} d + d^{2} \cdot 1 + 2 d a + 2 d b + 2 d c + 2 dd + 2 d \cdot 1 + 1 \cdot a + 1 \cdot b + 1 \cdot c + 1 \cdot d + 1 \cdot 1 : a^{3} + 3 a^{2} b + 3 a^{2} c + 3 a^{2} d + 3 a^{2} + 3 a b^{2} + 3 b^{2} c + 3 b^{2} d + 6 ab + 3 a c^{2} + 6 a c + 3 a d^{2} + 6 a d + 6 ab c + 6 ab d + 6 a c d + 3 a + b^{3} + 3 b^{2} + 3 b c^{2} + 6 b c + 3 b d^{2} + 6 b d + 6 b c d + 3 b + c^{3} + 3 c^{2} d + 3 c^{2} + 3 c d^{2} + 6 c d + 3 c + d^{3} + 3 d^{2} + 3 d + 1

= a^{3} + 3 a^{2} b + 3 a^{2} c + 3 a^{2} d + 3 a^{2} + 3 a b^{2} + 3 b^{2} c + 3 b^{2} d + 6 ab + 3 a c^{2} + 6 a c + 3 a d^{2} + 6 a d + 6 ab c + 6 ab d + 6 a c d + 3 a + b^{3} + 3 b^{2} + 3 b c^{2} + 6 b c + 3 b d^{2} + 6 b d + 6 b c d + 3 b + c^{3} + 3 c^{2} d + 3 c^{2} + 3 c d^{2} + 6 c d + 3 c + d^{3} + 3 d^{2} + 3 d + 1

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Frequently Asked Questions (FAQ)

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The solution to expand (a+b+c+d+1)^3 is