Solve h(u)=(3u^2+5)^3(3u-1)^2

$$h ( u ) = ( 3 u ^ { 2 } + 5 ) ^ { 3 } ( 3 u - 1 ) ^ { 2 }$$

Evaluate

$$\left(3u-1\right)^{2}\left(3u^{2}+5\right)^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(3u^{2}+5\right)^{3}.

$$\left(27\left(u^{2}\right)^{3}+135\left(u^{2}\right)^{2}+225u^{2}+125\right)\left(3u-1\right)^{2}$$

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

$$\left(27u^{6}+135\left(u^{2}\right)^{2}+225u^{2}+125\right)\left(3u-1\right)^{2}$$

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

$$\left(27u^{6}+135u^{4}+225u^{2}+125\right)\left(3u-1\right)^{2}$$

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

$$\left(27u^{6}+135u^{4}+225u^{2}+125\right)\left(9u^{2}-6u+1\right)$$

Use the distributive property to multiply 27u^{6}+135u^{4}+225u^{2}+125 by 9u^{2}-6u+1 and combine like terms.

$$243u^{8}-162u^{7}+1242u^{6}-810u^{5}+2160u^{4}-1350u^{3}+1350u^{2}-750u+125$$

Expand

$$243u^{8}-162u^{7}+1242u^{6}-810u^{5}+2160u^{4}-1350u^{3}+1350u^{2}-750u+125$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(3u^{2}+5\right)^{3}.

$$\left(27\left(u^{2}\right)^{3}+135\left(u^{2}\right)^{2}+225u^{2}+125\right)\left(3u-1\right)^{2}$$

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

$$\left(27u^{6}+135\left(u^{2}\right)^{2}+225u^{2}+125\right)\left(3u-1\right)^{2}$$

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

$$\left(27u^{6}+135u^{4}+225u^{2}+125\right)\left(3u-1\right)^{2}$$

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

$$\left(27u^{6}+135u^{4}+225u^{2}+125\right)\left(9u^{2}-6u+1\right)$$

Use the distributive property to multiply 27u^{6}+135u^{4}+225u^{2}+125 by 9u^{2}-6u+1 and combine like terms.

$$243u^{8}-162u^{7}+1242u^{6}-810u^{5}+2160u^{4}-1350u^{3}+1350u^{2}-750u+125$$