Solve (12x^{3}-6x^2)^2+(-6x^2+1)(2x^3)

$$( 12 x ^ { 3 } - 6 x ^ { 2 } ) ^ { 2 } + ( - 6 x ^ { 2 } + 1 ) ( 2 x ^ { 3 } )$$

Evaluate

$$2x^{3}\left(72x^{3}-78x^{2}+18x+1\right)$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12x^{3}-6x^{2}\right)^{2}.

$$144\left(x^{3}\right)^{2}-144x^{3}x^{2}+36\left(x^{2}\right)^{2}+\left(-6x^{2}+1\right)\times 2x^{3}$$

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

$$144x^{6}-144x^{3}x^{2}+36\left(x^{2}\right)^{2}+\left(-6x^{2}+1\right)\times 2x^{3}$$

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

$$144x^{6}-144x^{5}+36\left(x^{2}\right)^{2}+\left(-6x^{2}+1\right)\times 2x^{3}$$

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

$$144x^{6}-144x^{5}+36x^{4}+\left(-6x^{2}+1\right)\times 2x^{3}$$

Use the distributive property to multiply -6x^{2}+1 by 2.

$$144x^{6}-144x^{5}+36x^{4}+\left(-12x^{2}+2\right)x^{3}$$

Use the distributive property to multiply -12x^{2}+2 by x^{3}.

$$144x^{6}-144x^{5}+36x^{4}-12x^{5}+2x^{3}$$

Combine -144x^{5} and -12x^{5} to get -156x^{5}.

$$144x^{6}-156x^{5}+36x^{4}+2x^{3}$$

Expand

$$144x^{6}-156x^{5}+36x^{4}+2x^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12x^{3}-6x^{2}\right)^{2}.

$$144\left(x^{3}\right)^{2}-144x^{3}x^{2}+36\left(x^{2}\right)^{2}+\left(-6x^{2}+1\right)\times 2x^{3}$$

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

$$144x^{6}-144x^{3}x^{2}+36\left(x^{2}\right)^{2}+\left(-6x^{2}+1\right)\times 2x^{3}$$

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

$$144x^{6}-144x^{5}+36\left(x^{2}\right)^{2}+\left(-6x^{2}+1\right)\times 2x^{3}$$

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

$$144x^{6}-144x^{5}+36x^{4}+\left(-6x^{2}+1\right)\times 2x^{3}$$

Use the distributive property to multiply -6x^{2}+1 by 2.

$$144x^{6}-144x^{5}+36x^{4}+\left(-12x^{2}+2\right)x^{3}$$

Use the distributive property to multiply -12x^{2}+2 by x^{3}.

$$144x^{6}-144x^{5}+36x^{4}-12x^{5}+2x^{3}$$

Combine -144x^{5} and -12x^{5} to get -156x^{5}.

$$144x^{6}-156x^{5}+36x^{4}+2x^{3}$$