Solve (4-6x)^3-4(4-6x)^2+7(4-6x)-1

$$( 4 - 6 x ) ^ { 3 } - 4 ( 4 - 6 x ) ^ { 2 } + 7 ( 4 - 6 x ) - 1$$

Evaluate

$$27-138x+288x^{2}-216x^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(4-6x\right)^{3}.

$$64-288x+432x^{2}-216x^{3}-4\left(4-6x\right)^{2}+7\left(4-6x\right)-1$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-6x\right)^{2}.

$$64-288x+432x^{2}-216x^{3}-4\left(16-48x+36x^{2}\right)+7\left(4-6x\right)-1$$

Use the distributive property to multiply -4 by 16-48x+36x^{2}.

$$64-288x+432x^{2}-216x^{3}-64+192x-144x^{2}+7\left(4-6x\right)-1$$

Subtract 64 from 64 to get 0.

$$-288x+432x^{2}-216x^{3}+192x-144x^{2}+7\left(4-6x\right)-1$$

Combine -288x and 192x to get -96x.

$$-96x+432x^{2}-216x^{3}-144x^{2}+7\left(4-6x\right)-1$$

Combine 432x^{2} and -144x^{2} to get 288x^{2}.

$$-96x+288x^{2}-216x^{3}+7\left(4-6x\right)-1$$

Use the distributive property to multiply 7 by 4-6x.

$$-96x+288x^{2}-216x^{3}+28-42x-1$$

Combine -96x and -42x to get -138x.

$$-138x+288x^{2}-216x^{3}+28-1$$

Subtract 1 from 28 to get 27.

$$-138x+288x^{2}-216x^{3}+27$$

Expand

$$27-138x+288x^{2}-216x^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(4-6x\right)^{3}.

$$64-288x+432x^{2}-216x^{3}-4\left(4-6x\right)^{2}+7\left(4-6x\right)-1$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-6x\right)^{2}.

$$64-288x+432x^{2}-216x^{3}-4\left(16-48x+36x^{2}\right)+7\left(4-6x\right)-1$$

Use the distributive property to multiply -4 by 16-48x+36x^{2}.

$$64-288x+432x^{2}-216x^{3}-64+192x-144x^{2}+7\left(4-6x\right)-1$$

Subtract 64 from 64 to get 0.

$$-288x+432x^{2}-216x^{3}+192x-144x^{2}+7\left(4-6x\right)-1$$

Combine -288x and 192x to get -96x.

$$-96x+432x^{2}-216x^{3}-144x^{2}+7\left(4-6x\right)-1$$

Combine 432x^{2} and -144x^{2} to get 288x^{2}.

$$-96x+288x^{2}-216x^{3}+7\left(4-6x\right)-1$$

Use the distributive property to multiply 7 by 4-6x.

$$-96x+288x^{2}-216x^{3}+28-42x-1$$

Combine -96x and -42x to get -138x.

$$-138x+288x^{2}-216x^{3}+28-1$$

Subtract 1 from 28 to get 27.

$$-138x+288x^{2}-216x^{3}+27$$