Solve (x^2+1)^3-(x^3+1)^2

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

Evaluate

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

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x^{2}+1\right)^{3}.

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

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

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{3}+1\right)^{2}.

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

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

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

To find the opposite of x^{6}+2x^{3}+1, find the opposite of each term.

$$x^{6}+3x^{4}+3x^{2}+1-x^{6}-2x^{3}-1$$

Combine x^{6} and -x^{6} to get 0.

$$3x^{4}+3x^{2}+1-2x^{3}-1$$

Subtract 1 from 1 to get 0.

$$3x^{4}+3x^{2}-2x^{3}$$

Expand

$$3x^{4}-2x^{3}+3x^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x^{2}+1\right)^{3}.

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

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

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{3}+1\right)^{2}.

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

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

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

To find the opposite of x^{6}+2x^{3}+1, find the opposite of each term.

$$x^{6}+3x^{4}+3x^{2}+1-x^{6}-2x^{3}-1$$

Combine x^{6} and -x^{6} to get 0.

$$3x^{4}+3x^{2}+1-2x^{3}-1$$

Subtract 1 from 1 to get 0.

$$3x^{4}+3x^{2}-2x^{3}$$