Solve =(4x-1)^3-(5-x)^2-(x-1)^2

$$= ( 4 x - 1 ) ^ { 3 } - ( 5 - x ) ^ { 2 } - ( x - 1 ) ^ { 2 }$$

Evaluate

$$64x^{3}-50x^{2}+24x-27$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(4x-1\right)^{3}.

$$64x^{3}-48x^{2}+12x-1-\left(5-x\right)^{2}-\left(x-1\right)^{2}$$

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

$$64x^{3}-48x^{2}+12x-1-\left(25-10x+x^{2}\right)-\left(x-1\right)^{2}$$

To find the opposite of 25-10x+x^{2}, find the opposite of each term.

$$64x^{3}-48x^{2}+12x-1-25+10x-x^{2}-\left(x-1\right)^{2}$$

Subtract 25 from -1 to get -26.

$$64x^{3}-48x^{2}+12x-26+10x-x^{2}-\left(x-1\right)^{2}$$

Combine 12x and 10x to get 22x.

$$64x^{3}-48x^{2}+22x-26-x^{2}-\left(x-1\right)^{2}$$

Combine -48x^{2} and -x^{2} to get -49x^{2}.

$$64x^{3}-49x^{2}+22x-26-\left(x-1\right)^{2}$$

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

$$64x^{3}-49x^{2}+22x-26-\left(x^{2}-2x+1\right)$$

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

$$64x^{3}-49x^{2}+22x-26-x^{2}+2x-1$$

Combine -49x^{2} and -x^{2} to get -50x^{2}.

$$64x^{3}-50x^{2}+22x-26+2x-1$$

Combine 22x and 2x to get 24x.

$$64x^{3}-50x^{2}+24x-26-1$$

Subtract 1 from -26 to get -27.

$$64x^{3}-50x^{2}+24x-27$$

Expand

$$64x^{3}-50x^{2}+24x-27$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(4x-1\right)^{3}.

$$64x^{3}-48x^{2}+12x-1-\left(5-x\right)^{2}-\left(x-1\right)^{2}$$

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

$$64x^{3}-48x^{2}+12x-1-\left(25-10x+x^{2}\right)-\left(x-1\right)^{2}$$

To find the opposite of 25-10x+x^{2}, find the opposite of each term.

$$64x^{3}-48x^{2}+12x-1-25+10x-x^{2}-\left(x-1\right)^{2}$$

Subtract 25 from -1 to get -26.

$$64x^{3}-48x^{2}+12x-26+10x-x^{2}-\left(x-1\right)^{2}$$

Combine 12x and 10x to get 22x.

$$64x^{3}-48x^{2}+22x-26-x^{2}-\left(x-1\right)^{2}$$

Combine -48x^{2} and -x^{2} to get -49x^{2}.

$$64x^{3}-49x^{2}+22x-26-\left(x-1\right)^{2}$$

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

$$64x^{3}-49x^{2}+22x-26-\left(x^{2}-2x+1\right)$$

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

$$64x^{3}-49x^{2}+22x-26-x^{2}+2x-1$$

Combine -49x^{2} and -x^{2} to get -50x^{2}.

$$64x^{3}-50x^{2}+22x-26+2x-1$$

Combine 22x and 2x to get 24x.

$$64x^{3}-50x^{2}+24x-26-1$$

Subtract 1 from -26 to get -27.

$$64x^{3}-50x^{2}+24x-27$$