Solve 3left(4x^5-2xright)^2-9left(4x^5-2xright)-1

$$3 { \left(4 { x }^{ 5 } -2x \right) }^{ 2 } -9 \left( 4 { x }^{ 5 } -2x \right) -1$$

Evaluate

$$12\left(x\left(2x^{4}-1\right)\right)^{2}-36x^{5}+18x-1$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x^{5}-2x\right)^{2}.

$$3\left(16\left(x^{5}\right)^{2}-16x^{5}x+4x^{2}\right)-9\left(4x^{5}-2x\right)-1$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$3\left(16x^{10}-16x^{5}x+4x^{2}\right)-9\left(4x^{5}-2x\right)-1$$

To multiply powers of the same base, add their exponents. Add 5 and 1 to get 6.

$$3\left(16x^{10}-16x^{6}+4x^{2}\right)-9\left(4x^{5}-2x\right)-1$$

Use the distributive property to multiply 3 by 16x^{10}-16x^{6}+4x^{2}.

$$48x^{10}-48x^{6}+12x^{2}-9\left(4x^{5}-2x\right)-1$$

Use the distributive property to multiply -9 by 4x^{5}-2x.

$$48x^{10}-48x^{6}+12x^{2}-36x^{5}+18x-1$$

Expand

$$48x^{10}-48x^{6}-36x^{5}+12x^{2}+18x-1$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x^{5}-2x\right)^{2}.

$$3\left(16\left(x^{5}\right)^{2}-16x^{5}x+4x^{2}\right)-9\left(4x^{5}-2x\right)-1$$

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$3\left(16x^{10}-16x^{5}x+4x^{2}\right)-9\left(4x^{5}-2x\right)-1$$

To multiply powers of the same base, add their exponents. Add 5 and 1 to get 6.

$$3\left(16x^{10}-16x^{6}+4x^{2}\right)-9\left(4x^{5}-2x\right)-1$$

Use the distributive property to multiply 3 by 16x^{10}-16x^{6}+4x^{2}.

$$48x^{10}-48x^{6}+12x^{2}-9\left(4x^{5}-2x\right)-1$$

Use the distributive property to multiply -9 by 4x^{5}-2x.

$$48x^{10}-48x^{6}+12x^{2}-36x^{5}+18x-1$$