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Popular >

expand (1+x)^5(1+ax)^6

Solution

expand

(1 + x)^{5} (1 + a x)^{6}

Solution

1 + 6 a x + 15 a^{2} x^{2} + 20 a^{3} x^{3} + 15 a^{4} x^{4} + 6 a^{5} x^{5} + a^{6} x^{6} + 5 x + 30 a x^{2} + 75 x^{3} a^{2} + 100 x^{4} a^{3} + 75 x^{5} a^{4} + 30 x^{6} a^{5} + 5 x^{7} a^{6} + 10 x^{2} + 60 a x^{3} + 150 x^{4} a^{2} + 200 x^{5} a^{3} + 150 x^{6} a^{4} + 60 x^{7} a^{5} + 10 x^{8} a^{6} + 10 x^{3} + 60 a x^{4} + 150 x^{5} a^{2} + 200 x^{6} a^{3} + 150 x^{7} a^{4} + 60 x^{8} a^{5} + 10 x^{9} a^{6} + 5 x^{4} + 30 a x^{5} + 75 x^{6} a^{2} + 100 x^{7} a^{3} + 75 x^{8} a^{4} + 30 x^{9} a^{5} + 5 x^{10} a^{6} + x^{5} + 6 a x^{6} + 15 x^{7} a^{2} + 20 x^{8} a^{3} + 15 x^{9} a^{4} + 6 x^{10} a^{5} + x^{11} a^{6}

Solution steps

(1 + x)^{5} (1 + a x)^{6}

Expand

(1 + x)^{5} : 1 + 5 x + 10 x^{2} + 10 x^{3} + 5 x^{4} + x^{5}

= (1 + 5 x + 10 x^{2} + 10 x^{3} + 5 x^{4} + x^{5}) (1 + a x)^{6}

= (1 + a x)^{6} (1 + 5 x + 10 x^{2} + 10 x^{3} + 5 x^{4} + x^{5})

Distribute parentheses

= (1 + a x)^{6} \cdot 1 + (1 + a x)^{6} \cdot 5 x + (1 + a x)^{6} \cdot 10 x^{2} + (1 + a x)^{6} \cdot 10 x^{3} + (1 + a x)^{6} \cdot 5 x^{4} + (1 + a x)^{6} x^{5}

Simplify

(1 + a x)^{6} \cdot 1 + (1 + a x)^{6} \cdot 5 x + (1 + a x)^{6} \cdot 10 x^{2} + (1 + a x)^{6} \cdot 10 x^{3} + (1 + a x)^{6} \cdot 5 x^{4} + (1 + a x)^{6} x^{5} : (1 + a x)^{6} + 5 x (1 + a x)^{6} + 10 x^{2} (1 + a x)^{6} + 10 x^{3} (1 + a x)^{6} + 5 x^{4} (1 + a x)^{6} + x^{5} (1 + a x)^{6}

= (1 + a x)^{6} + 5 x (1 + a x)^{6} + 10 x^{2} (1 + a x)^{6} + 10 x^{3} (1 + a x)^{6} + 5 x^{4} (1 + a x)^{6} + x^{5} (1 + a x)^{6}

Expand

(1 + a x)^{6} : 1 + 6 a x + 15 a^{2} x^{2} + 20 a^{3} x^{3} + 15 a^{4} x^{4} + 6 a^{5} x^{5} + a^{6} x^{6}

= 1 + 6 a x + 15 a^{2} x^{2} + 20 a^{3} x^{3} + 15 a^{4} x^{4} + 6 a^{5} x^{5} + a^{6} x^{6} + 5 x (1 + a x)^{6} + 10 x^{2} (1 + a x)^{6} + 10 x^{3} (1 + a x)^{6} + 5 x^{4} (1 + a x)^{6} + x^{5} (1 + a x)^{6}

Expand

5 x (1 + a x)^{6} : 5 x + 30 a x^{2} + 75 x^{3} a^{2} + 100 x^{4} a^{3} + 75 x^{5} a^{4} + 30 x^{6} a^{5} + 5 x^{7} a^{6}

= 1 + 6 a x + 15 a^{2} x^{2} + 20 a^{3} x^{3} + 15 a^{4} x^{4} + 6 a^{5} x^{5} + a^{6} x^{6} + 5 x + 30 a x^{2} + 75 x^{3} a^{2} + 100 x^{4} a^{3} + 75 x^{5} a^{4} + 30 x^{6} a^{5} + 5 x^{7} a^{6} + 10 x^{2} (1 + a x)^{6} + 10 x^{3} (1 + a x)^{6} + 5 x^{4} (1 + a x)^{6} + x^{5} (1 + a x)^{6}

Expand

10 x^{2} (1 + a x)^{6} : 10 x^{2} + 60 a x^{3} + 150 x^{4} a^{2} + 200 x^{5} a^{3} + 150 x^{6} a^{4} + 60 x^{7} a^{5} + 10 x^{8} a^{6}

= 1 + 6 a x + 15 a^{2} x^{2} + 20 a^{3} x^{3} + 15 a^{4} x^{4} + 6 a^{5} x^{5} + a^{6} x^{6} + 5 x + 30 a x^{2} + 75 x^{3} a^{2} + 100 x^{4} a^{3} + 75 x^{5} a^{4} + 30 x^{6} a^{5} + 5 x^{7} a^{6} + 10 x^{2} + 60 a x^{3} + 150 x^{4} a^{2} + 200 x^{5} a^{3} + 150 x^{6} a^{4} + 60 x^{7} a^{5} + 10 x^{8} a^{6} + 10 x^{3} (1 + a x)^{6} + 5 x^{4} (1 + a x)^{6} + x^{5} (1 + a x)^{6}

Expand

10 x^{3} (1 + a x)^{6} : 10 x^{3} + 60 a x^{4} + 150 x^{5} a^{2} + 200 x^{6} a^{3} + 150 x^{7} a^{4} + 60 x^{8} a^{5} + 10 x^{9} a^{6}

= 1 + 6 a x + 15 a^{2} x^{2} + 20 a^{3} x^{3} + 15 a^{4} x^{4} + 6 a^{5} x^{5} + a^{6} x^{6} + 5 x + 30 a x^{2} + 75 x^{3} a^{2} + 100 x^{4} a^{3} + 75 x^{5} a^{4} + 30 x^{6} a^{5} + 5 x^{7} a^{6} + 10 x^{2} + 60 a x^{3} + 150 x^{4} a^{2} + 200 x^{5} a^{3} + 150 x^{6} a^{4} + 60 x^{7} a^{5} + 10 x^{8} a^{6} + 10 x^{3} + 60 a x^{4} + 150 x^{5} a^{2} + 200 x^{6} a^{3} + 150 x^{7} a^{4} + 60 x^{8} a^{5} + 10 x^{9} a^{6} + 5 x^{4} (1 + a x)^{6} + x^{5} (1 + a x)^{6}

Expand

5 x^{4} (1 + a x)^{6} : 5 x^{4} + 30 a x^{5} + 75 x^{6} a^{2} + 100 x^{7} a^{3} + 75 x^{8} a^{4} + 30 x^{9} a^{5} + 5 x^{10} a^{6}

= 1 + 6 a x + 15 a^{2} x^{2} + 20 a^{3} x^{3} + 15 a^{4} x^{4} + 6 a^{5} x^{5} + a^{6} x^{6} + 5 x + 30 a x^{2} + 75 x^{3} a^{2} + 100 x^{4} a^{3} + 75 x^{5} a^{4} + 30 x^{6} a^{5} + 5 x^{7} a^{6} + 10 x^{2} + 60 a x^{3} + 150 x^{4} a^{2} + 200 x^{5} a^{3} + 150 x^{6} a^{4} + 60 x^{7} a^{5} + 10 x^{8} a^{6} + 10 x^{3} + 60 a x^{4} + 150 x^{5} a^{2} + 200 x^{6} a^{3} + 150 x^{7} a^{4} + 60 x^{8} a^{5} + 10 x^{9} a^{6} + 5 x^{4} + 30 a x^{5} + 75 x^{6} a^{2} + 100 x^{7} a^{3} + 75 x^{8} a^{4} + 30 x^{9} a^{5} + 5 x^{10} a^{6} + x^{5} (1 + a x)^{6}

Expand

x^{5} (1 + a x)^{6} : x^{5} + 6 a x^{6} + 15 x^{7} a^{2} + 20 x^{8} a^{3} + 15 x^{9} a^{4} + 6 x^{10} a^{5} + x^{11} a^{6}

= 1 + 6 a x + 15 a^{2} x^{2} + 20 a^{3} x^{3} + 15 a^{4} x^{4} + 6 a^{5} x^{5} + a^{6} x^{6} + 5 x + 30 a x^{2} + 75 x^{3} a^{2} + 100 x^{4} a^{3} + 75 x^{5} a^{4} + 30 x^{6} a^{5} + 5 x^{7} a^{6} + 10 x^{2} + 60 a x^{3} + 150 x^{4} a^{2} + 200 x^{5} a^{3} + 150 x^{6} a^{4} + 60 x^{7} a^{5} + 10 x^{8} a^{6} + 10 x^{3} + 60 a x^{4} + 150 x^{5} a^{2} + 200 x^{6} a^{3} + 150 x^{7} a^{4} + 60 x^{8} a^{5} + 10 x^{9} a^{6} + 5 x^{4} + 30 a x^{5} + 75 x^{6} a^{2} + 100 x^{7} a^{3} + 75 x^{8} a^{4} + 30 x^{9} a^{5} + 5 x^{10} a^{6} + x^{5} + 6 a x^{6} + 15 x^{7} a^{2} + 20 x^{8} a^{3} + 15 x^{9} a^{4} + 6 x^{10} a^{5} + x^{11} a^{6}

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Frequently Asked Questions (FAQ)

What is expand (1+x)^5(1+ax)^6 ?
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