Solve 2k^{4}-(k^3-1)^2+4(k+4-2k^2)^2

$$2 k ^ { 4 } - ( k ^ { 3 } - 1 ) ^ { 2 } + 4 ( k + 4 - 2 k ^ { 2 } ) ^ { 2 }$$

Evaluate

$$63+32k-60k^{2}-14k^{3}+18k^{4}-k^{6}$$

Solution Steps

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

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

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

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

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

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

Square k+4-2k^{2}.

$$2k^{4}-k^{6}+2k^{3}-1+4\left(4k^{4}-4k^{3}-15k^{2}+8k+16\right)$$

Use the distributive property to multiply 4 by 4k^{4}-4k^{3}-15k^{2}+8k+16.

$$2k^{4}-k^{6}+2k^{3}-1+16k^{4}-16k^{3}-60k^{2}+32k+64$$

Combine 2k^{4} and 16k^{4} to get 18k^{4}.

$$18k^{4}-k^{6}+2k^{3}-1-16k^{3}-60k^{2}+32k+64$$

Combine 2k^{3} and -16k^{3} to get -14k^{3}.

$$18k^{4}-k^{6}-14k^{3}-1-60k^{2}+32k+64$$

Add -1 and 64 to get 63.

$$18k^{4}-k^{6}-14k^{3}+63-60k^{2}+32k$$

Expand

$$63+32k-60k^{2}-14k^{3}+18k^{4}-k^{6}$$

Solution Steps

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

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

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

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

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

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

Square k+4-2k^{2}.

$$2k^{4}-k^{6}+2k^{3}-1+4\left(4k^{4}-4k^{3}-15k^{2}+8k+16\right)$$

Use the distributive property to multiply 4 by 4k^{4}-4k^{3}-15k^{2}+8k+16.

$$2k^{4}-k^{6}+2k^{3}-1+16k^{4}-16k^{3}-60k^{2}+32k+64$$

Combine 2k^{4} and 16k^{4} to get 18k^{4}.

$$18k^{4}-k^{6}+2k^{3}-1-16k^{3}-60k^{2}+32k+64$$

Combine 2k^{3} and -16k^{3} to get -14k^{3}.

$$18k^{4}-k^{6}-14k^{3}-1-60k^{2}+32k+64$$

Add -1 and 64 to get 63.

$$18k^{4}-k^{6}-14k^{3}+63-60k^{2}+32k$$