Solve left(x^2-3right)^3-2left(x^2-3right)^3-20

$${ \left( { x }^{ 2 } -3 \right) }^{ 3 } -2 { \left( { x }^{ 2 } -3 \right) }^{ 3 } -20$$

Evaluate

$$-\left(x^{2}-3\right)^{3}-20$$

Solution Steps

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

$$\left(x^{2}\right)^{3}-9\left(x^{2}\right)^{2}+27x^{2}-27-2\left(x^{2}-3\right)^{3}-20$$

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

$$x^{6}-9\left(x^{2}\right)^{2}+27x^{2}-27-2\left(x^{2}-3\right)^{3}-20$$

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

$$x^{6}-9x^{4}+27x^{2}-27-2\left(x^{2}-3\right)^{3}-20$$

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

$$x^{6}-9x^{4}+27x^{2}-27-2\left(\left(x^{2}\right)^{3}-9\left(x^{2}\right)^{2}+27x^{2}-27\right)-20$$

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

$$x^{6}-9x^{4}+27x^{2}-27-2\left(x^{6}-9\left(x^{2}\right)^{2}+27x^{2}-27\right)-20$$

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

$$x^{6}-9x^{4}+27x^{2}-27-2\left(x^{6}-9x^{4}+27x^{2}-27\right)-20$$

Use the distributive property to multiply -2 by x^{6}-9x^{4}+27x^{2}-27.

$$x^{6}-9x^{4}+27x^{2}-27-2x^{6}+18x^{4}-54x^{2}+54-20$$

Combine x^{6} and -2x^{6} to get -x^{6}.

$$-x^{6}-9x^{4}+27x^{2}-27+18x^{4}-54x^{2}+54-20$$

Combine -9x^{4} and 18x^{4} to get 9x^{4}.

$$-x^{6}+9x^{4}+27x^{2}-27-54x^{2}+54-20$$

Combine 27x^{2} and -54x^{2} to get -27x^{2}.

$$-x^{6}+9x^{4}-27x^{2}-27+54-20$$

Add -27 and 54 to get 27.

$$-x^{6}+9x^{4}-27x^{2}+27-20$$

Subtract 20 from 27 to get 7.

$$-x^{6}+9x^{4}-27x^{2}+7$$

Expand

$$7-27x^{2}+9x^{4}-x^{6}$$

Solution Steps

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

$$\left(x^{2}\right)^{3}-9\left(x^{2}\right)^{2}+27x^{2}-27-2\left(x^{2}-3\right)^{3}-20$$

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

$$x^{6}-9\left(x^{2}\right)^{2}+27x^{2}-27-2\left(x^{2}-3\right)^{3}-20$$

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

$$x^{6}-9x^{4}+27x^{2}-27-2\left(x^{2}-3\right)^{3}-20$$

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

$$x^{6}-9x^{4}+27x^{2}-27-2\left(\left(x^{2}\right)^{3}-9\left(x^{2}\right)^{2}+27x^{2}-27\right)-20$$

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

$$x^{6}-9x^{4}+27x^{2}-27-2\left(x^{6}-9\left(x^{2}\right)^{2}+27x^{2}-27\right)-20$$

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

$$x^{6}-9x^{4}+27x^{2}-27-2\left(x^{6}-9x^{4}+27x^{2}-27\right)-20$$

Use the distributive property to multiply -2 by x^{6}-9x^{4}+27x^{2}-27.

$$x^{6}-9x^{4}+27x^{2}-27-2x^{6}+18x^{4}-54x^{2}+54-20$$

Combine x^{6} and -2x^{6} to get -x^{6}.

$$-x^{6}-9x^{4}+27x^{2}-27+18x^{4}-54x^{2}+54-20$$

Combine -9x^{4} and 18x^{4} to get 9x^{4}.

$$-x^{6}+9x^{4}+27x^{2}-27-54x^{2}+54-20$$

Combine 27x^{2} and -54x^{2} to get -27x^{2}.

$$-x^{6}+9x^{4}-27x^{2}-27+54-20$$

Add -27 and 54 to get 27.

$$-x^{6}+9x^{4}-27x^{2}+27-20$$

Subtract 20 from 27 to get 7.

$$-x^{6}+9x^{4}-27x^{2}+7$$