expand (x+a)^4(x-a)^3

Solution

expand

(x + a)^{4} (x - a)^{3}

x^{7} - a^{7} + 19 x^{3} a^{4} + a x^{6} + 9 x^{5} a^{2} + 16 x^{4} a^{3} - 12 a^{2} x^{5} - 19 a^{3} x^{4} - 16 a^{4} x^{3} - 9 a^{5} x^{2} + 12 x^{2} a^{5} - x a^{6}

Solution steps

(x + a)^{4} (x - a)^{3}

Expand

(x - a)^{3} : x^{3} - 3 a x^{2} + 3 x a^{2} - a^{3}

= (x + a)^{4} (x^{3} - 3 a x^{2} + 3 x a^{2} - a^{3})

Distribute parentheses

= (x + a)^{4} x^{3} + (x + a)^{4} (- 3 a x^{2}) + (x + a)^{4} \cdot 3 x a^{2} + (x + a)^{4} (- a^{3})

Simplify

(x + a)^{4} x^{3} + (x + a)^{4} (- 3 a x^{2}) + (x + a)^{4} \cdot 3 x a^{2} + (x + a)^{4} (- a^{3}) : x^{3} (x + a)^{4} - 3 a x^{2} (x + a)^{4} + 3 x a^{2} (x + a)^{4} - a^{3} (x + a)^{4}

= x^{3} (x + a)^{4} - 3 a x^{2} (x + a)^{4} + 3 x a^{2} (x + a)^{4} - a^{3} (x + a)^{4}

Expand

x^{3} (x + a)^{4} : x^{7} + 4 a x^{6} + 6 x^{5} a^{2} + 4 x^{4} a^{3} + x^{3} a^{4}

= x^{7} + 4 a x^{6} + 6 x^{5} a^{2} + 4 x^{4} a^{3} + x^{3} a^{4} - 3 a x^{2} (x + a)^{4} + 3 x a^{2} (x + a)^{4} - a^{3} (x + a)^{4}

Expand

- 3 a x^{2} (x + a)^{4} : - 3 a x^{6} - 12 a^{2} x^{5} - 18 a^{3} x^{4} - 12 a^{4} x^{3} - 3 a^{5} x^{2}

= x^{7} + 4 a x^{6} + 6 x^{5} a^{2} + 4 x^{4} a^{3} + x^{3} a^{4} - 3 a x^{6} - 12 a^{2} x^{5} - 18 a^{3} x^{4} - 12 a^{4} x^{3} - 3 a^{5} x^{2} + 3 x a^{2} (x + a)^{4} - a^{3} (x + a)^{4}

Expand

3 x a^{2} (x + a)^{4} : 3 x^{5} a^{2} + 12 x^{4} a^{3} + 18 x^{3} a^{4} + 12 x^{2} a^{5} + 3 x a^{6}

= x^{7} + 4 a x^{6} + 6 x^{5} a^{2} + 4 x^{4} a^{3} + x^{3} a^{4} - 3 a x^{6} - 12 a^{2} x^{5} - 18 a^{3} x^{4} - 12 a^{4} x^{3} - 3 a^{5} x^{2} + 3 x^{5} a^{2} + 12 x^{4} a^{3} + 18 x^{3} a^{4} + 12 x^{2} a^{5} + 3 x a^{6} - a^{3} (x + a)^{4}

Expand

- a^{3} (x + a)^{4} : - a^{3} x^{4} - 4 a^{4} x^{3} - 6 a^{5} x^{2} - 4 x a^{6} - a^{7}

= x^{7} + 4 a x^{6} + 6 x^{5} a^{2} + 4 x^{4} a^{3} + x^{3} a^{4} - 3 a x^{6} - 12 a^{2} x^{5} - 18 a^{3} x^{4} - 12 a^{4} x^{3} - 3 a^{5} x^{2} + 3 x^{5} a^{2} + 12 x^{4} a^{3} + 18 x^{3} a^{4} + 12 x^{2} a^{5} + 3 x a^{6} - a^{3} x^{4} - 4 a^{4} x^{3} - 6 a^{5} x^{2} - 4 x a^{6} - a^{7}

Simplify

x^{7} + 4 a x^{6} + 6 x^{5} a^{2} + 4 x^{4} a^{3} + x^{3} a^{4} - 3 a x^{6} - 12 a^{2} x^{5} - 18 a^{3} x^{4} - 12 a^{4} x^{3} - 3 a^{5} x^{2} + 3 x^{5} a^{2} + 12 x^{4} a^{3} + 18 x^{3} a^{4} + 12 x^{2} a^{5} + 3 x a^{6} - a^{3} x^{4} - 4 a^{4} x^{3} - 6 a^{5} x^{2} - 4 x a^{6} - a^{7} : x^{7} - a^{7} + 19 x^{3} a^{4} + a x^{6} + 9 x^{5} a^{2} + 16 x^{4} a^{3} - 12 a^{2} x^{5} - 19 a^{3} x^{4} - 16 a^{4} x^{3} - 9 a^{5} x^{2} + 12 x^{2} a^{5} - x a^{6}

= x^{7} - a^{7} + 19 x^{3} a^{4} + a x^{6} + 9 x^{5} a^{2} + 16 x^{4} a^{3} - 12 a^{2} x^{5} - 19 a^{3} x^{4} - 16 a^{4} x^{3} - 9 a^{5} x^{2} + 12 x^{2} a^{5} - x a^{6}