Solve (y-3)(y^2+9)(y+3)-(2y^2-y)^2-19

$$( y - 3 ) ( y ^ { 2 } + 9 ) ( y + 3 ) - ( 2 y ^ { 2 } - y ) ^ { 2 } - 19$$

Evaluate

$$-\left(y\left(2y-1\right)\right)^{2}+y^{4}-100$$

Solution Steps

Use the distributive property to multiply y-3 by y^{2}+9.

$$\left(y^{3}+9y-3y^{2}-27\right)\left(y+3\right)-\left(2y^{2}-y\right)^{2}-19$$

Use the distributive property to multiply y^{3}+9y-3y^{2}-27 by y+3 and combine like terms.

$$y^{4}-81-\left(2y^{2}-y\right)^{2}-19$$

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

$$y^{4}-81-\left(4\left(y^{2}\right)^{2}-4y^{2}y+y^{2}\right)-19$$

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

$$y^{4}-81-\left(4y^{4}-4y^{2}y+y^{2}\right)-19$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$y^{4}-81-\left(4y^{4}-4y^{3}+y^{2}\right)-19$$

To find the opposite of 4y^{4}-4y^{3}+y^{2}, find the opposite of each term.

$$y^{4}-81-4y^{4}+4y^{3}-y^{2}-19$$

Combine y^{4} and -4y^{4} to get -3y^{4}.

$$-3y^{4}-81+4y^{3}-y^{2}-19$$

Subtract 19 from -81 to get -100.

$$-3y^{4}-100+4y^{3}-y^{2}$$

Expand

$$-3y^{4}+4y^{3}-y^{2}-100$$

Solution Steps

Use the distributive property to multiply y-3 by y^{2}+9.

$$\left(y^{3}+9y-3y^{2}-27\right)\left(y+3\right)-\left(2y^{2}-y\right)^{2}-19$$

Use the distributive property to multiply y^{3}+9y-3y^{2}-27 by y+3 and combine like terms.

$$y^{4}-81-\left(2y^{2}-y\right)^{2}-19$$

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

$$y^{4}-81-\left(4\left(y^{2}\right)^{2}-4y^{2}y+y^{2}\right)-19$$

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

$$y^{4}-81-\left(4y^{4}-4y^{2}y+y^{2}\right)-19$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$y^{4}-81-\left(4y^{4}-4y^{3}+y^{2}\right)-19$$

To find the opposite of 4y^{4}-4y^{3}+y^{2}, find the opposite of each term.

$$y^{4}-81-4y^{4}+4y^{3}-y^{2}-19$$

Combine y^{4} and -4y^{4} to get -3y^{4}.

$$-3y^{4}-81+4y^{3}-y^{2}-19$$

Subtract 19 from -81 to get -100.

$$-3y^{4}-100+4y^{3}-y^{2}$$