Solve (x^{4}-2x)(x^2+1)-(x^2-3y)(x^2+3y)+2x-9y^2

$$( x ^ { 4 } - 2 x ) ( x ^ { 2 } + 1 ) - ( x ^ { 2 } - 3 y ) ( x ^ { 2 } + 3 y ) + 2 x - 9 y ^ { 2 }$$

Evaluate

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

Solution Steps

Use the distributive property to multiply x^{4}-2x by x^{2}+1.

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

Consider \left(x^{2}-3y\right)\left(x^{2}+3y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Expand \left(3y\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

To find the opposite of x^{4}-9y^{2}, find the opposite of each term.

$$x^{6}+x^{4}-2x^{3}-2x-x^{4}+9y^{2}+2x-9y^{2}$$

Combine x^{4} and -x^{4} to get 0.

$$x^{6}-2x^{3}-2x+9y^{2}+2x-9y^{2}$$

Combine -2x and 2x to get 0.

$$x^{6}-2x^{3}+9y^{2}-9y^{2}$$

Combine 9y^{2} and -9y^{2} to get 0.

$$x^{6}-2x^{3}$$

Expand

$$x^{6}-2x^{3}$$

Solution Steps

Use the distributive property to multiply x^{4}-2x by x^{2}+1.

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

Consider \left(x^{2}-3y\right)\left(x^{2}+3y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Expand \left(3y\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

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

To find the opposite of x^{4}-9y^{2}, find the opposite of each term.

$$x^{6}+x^{4}-2x^{3}-2x-x^{4}+9y^{2}+2x-9y^{2}$$

Combine x^{4} and -x^{4} to get 0.

$$x^{6}-2x^{3}-2x+9y^{2}+2x-9y^{2}$$

Combine -2x and 2x to get 0.

$$x^{6}-2x^{3}+9y^{2}-9y^{2}$$

Combine 9y^{2} and -9y^{2} to get 0.

$$x^{6}-2x^{3}$$