Solve g(x)=(3x^4+8)^2(x^3-7x)^3

$$g ( x ) = ( 3 x ^ { 4 } + 8 ) ^ { 2 } ( x ^ { 3 } - 7 x ) ^ { 3 }$$

Evaluate

$$\left(3x^{4}+8\right)^{2}\left(x\left(x^{2}-7\right)\right)^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x^{4}+8\right)^{2}.

$$\left(9\left(x^{4}\right)^{2}+48x^{4}+64\right)\left(x^{3}-7x\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\left(9x^{8}+48x^{4}+64\right)\left(x^{3}-7x\right)^{3}$$

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

$$\left(9x^{8}+48x^{4}+64\right)\left(\left(x^{3}\right)^{3}-21\left(x^{3}\right)^{2}x+147x^{3}x^{2}-343x^{3}\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\left(9x^{8}+48x^{4}+64\right)\left(x^{9}-21\left(x^{3}\right)^{2}x+147x^{3}x^{2}-343x^{3}\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\left(9x^{8}+48x^{4}+64\right)\left(x^{9}-21x^{6}x+147x^{3}x^{2}-343x^{3}\right)$$

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$\left(9x^{8}+48x^{4}+64\right)\left(x^{9}-21x^{7}+147x^{3}x^{2}-343x^{3}\right)$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\left(9x^{8}+48x^{4}+64\right)\left(x^{9}-21x^{7}+147x^{5}-343x^{3}\right)$$

Use the distributive property to multiply 9x^{8}+48x^{4}+64 by x^{9}-21x^{7}+147x^{5}-343x^{3} and combine like terms.

$$9x^{17}-189x^{15}+1371x^{13}-4095x^{11}+7120x^{9}-17808x^{7}+9408x^{5}-21952x^{3}$$

Expand

$$9x^{17}-189x^{15}+1371x^{13}-4095x^{11}+7120x^{9}-17808x^{7}+9408x^{5}-21952x^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x^{4}+8\right)^{2}.

$$\left(9\left(x^{4}\right)^{2}+48x^{4}+64\right)\left(x^{3}-7x\right)^{3}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\left(9x^{8}+48x^{4}+64\right)\left(x^{3}-7x\right)^{3}$$

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

$$\left(9x^{8}+48x^{4}+64\right)\left(\left(x^{3}\right)^{3}-21\left(x^{3}\right)^{2}x+147x^{3}x^{2}-343x^{3}\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\left(9x^{8}+48x^{4}+64\right)\left(x^{9}-21\left(x^{3}\right)^{2}x+147x^{3}x^{2}-343x^{3}\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\left(9x^{8}+48x^{4}+64\right)\left(x^{9}-21x^{6}x+147x^{3}x^{2}-343x^{3}\right)$$

To multiply powers of the same base, add their exponents. Add 6 and 1 to get 7.

$$\left(9x^{8}+48x^{4}+64\right)\left(x^{9}-21x^{7}+147x^{3}x^{2}-343x^{3}\right)$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\left(9x^{8}+48x^{4}+64\right)\left(x^{9}-21x^{7}+147x^{5}-343x^{3}\right)$$

Use the distributive property to multiply 9x^{8}+48x^{4}+64 by x^{9}-21x^{7}+147x^{5}-343x^{3} and combine like terms.

$$9x^{17}-189x^{15}+1371x^{13}-4095x^{11}+7120x^{9}-17808x^{7}+9408x^{5}-21952x^{3}$$