Solve 9(x-6)^2(x-13)^3(x+7)^3

$$9 ( x - 6 ) ^ { 2 } ( x - 13 ) ^ { 3 } ( x + 7 ) ^ { 3 }$$

Evaluate

$$9\left(x-6\right)^{2}\left(\left(x-13\right)\left(x+7\right)\right)^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$9\left(x^{2}-12x+36\right)\left(x-13\right)^{3}\left(x+7\right)^{3}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-13\right)^{3}.

$$9\left(x^{2}-12x+36\right)\left(x^{3}-39x^{2}+507x-2197\right)\left(x+7\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+7\right)^{3}.

$$9\left(x^{2}-12x+36\right)\left(x^{3}-39x^{2}+507x-2197\right)\left(x^{3}+21x^{2}+147x+343\right)$$

Use the distributive property to multiply 9 by x^{2}-12x+36.

$$\left(9x^{2}-108x+324\right)\left(x^{3}-39x^{2}+507x-2197\right)\left(x^{3}+21x^{2}+147x+343\right)$$

Use the distributive property to multiply 9x^{2}-108x+324 by x^{3}-39x^{2}+507x-2197 and combine like terms.

$$\left(9x^{5}-459x^{4}+9099x^{3}-87165x^{2}+401544x-711828\right)\left(x^{3}+21x^{2}+147x+343\right)$$

Use the distributive property to multiply 9x^{5}-459x^{4}+9099x^{3}-87165x^{2}+401544x-711828 by x^{3}+21x^{2}+147x+343 and combine like terms.

$$9x^{8}-270x^{7}+783x^{6}+39528x^{5}-248805x^{4}-1971702x^{3}[phone]x^{2}[phone]x-244157004$$

Expand

$$9x^{8}-270x^{7}+783x^{6}+39528x^{5}-248805x^{4}-1971702x^{3}[phone]x^{2}[phone]x-244157004$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$9\left(x^{2}-12x+36\right)\left(x-13\right)^{3}\left(x+7\right)^{3}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-13\right)^{3}.

$$9\left(x^{2}-12x+36\right)\left(x^{3}-39x^{2}+507x-2197\right)\left(x+7\right)^{3}$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+7\right)^{3}.

$$9\left(x^{2}-12x+36\right)\left(x^{3}-39x^{2}+507x-2197\right)\left(x^{3}+21x^{2}+147x+343\right)$$

Use the distributive property to multiply 9 by x^{2}-12x+36.

$$\left(9x^{2}-108x+324\right)\left(x^{3}-39x^{2}+507x-2197\right)\left(x^{3}+21x^{2}+147x+343\right)$$

Use the distributive property to multiply 9x^{2}-108x+324 by x^{3}-39x^{2}+507x-2197 and combine like terms.

$$\left(9x^{5}-459x^{4}+9099x^{3}-87165x^{2}+401544x-711828\right)\left(x^{3}+21x^{2}+147x+343\right)$$

Use the distributive property to multiply 9x^{5}-459x^{4}+9099x^{3}-87165x^{2}+401544x-711828 by x^{3}+21x^{2}+147x+343 and combine like terms.

$$9x^{8}-270x^{7}+783x^{6}+39528x^{5}-248805x^{4}-1971702x^{3}[phone]x^{2}[phone]x-244157004$$