Solve P(x-1)=5(x-1)^3-2(x-1)^2+x+1

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

Evaluate

$$5x^{3}-17x^{2}+20x-6$$

Solution Steps

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

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

Use the distributive property to multiply 5 by x^{3}-3x^{2}+3x-1.

$$5x^{3}-15x^{2}+15x-5-2\left(x-1\right)^{2}+x+1$$

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

$$5x^{3}-15x^{2}+15x-5-2\left(x^{2}-2x+1\right)+x+1$$

Use the distributive property to multiply -2 by x^{2}-2x+1.

$$5x^{3}-15x^{2}+15x-5-2x^{2}+4x-2+x+1$$

Combine -15x^{2} and -2x^{2} to get -17x^{2}.

$$5x^{3}-17x^{2}+15x-5+4x-2+x+1$$

Combine 15x and 4x to get 19x.

$$5x^{3}-17x^{2}+19x-5-2+x+1$$

Subtract 2 from -5 to get -7.

$$5x^{3}-17x^{2}+19x-7+x+1$$

Combine 19x and x to get 20x.

$$5x^{3}-17x^{2}+20x-7+1$$

Add -7 and 1 to get -6.

$$5x^{3}-17x^{2}+20x-6$$

Expand

$$5x^{3}-17x^{2}+20x-6$$

Solution Steps

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

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

Use the distributive property to multiply 5 by x^{3}-3x^{2}+3x-1.

$$5x^{3}-15x^{2}+15x-5-2\left(x-1\right)^{2}+x+1$$

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

$$5x^{3}-15x^{2}+15x-5-2\left(x^{2}-2x+1\right)+x+1$$

Use the distributive property to multiply -2 by x^{2}-2x+1.

$$5x^{3}-15x^{2}+15x-5-2x^{2}+4x-2+x+1$$

Combine -15x^{2} and -2x^{2} to get -17x^{2}.

$$5x^{3}-17x^{2}+15x-5+4x-2+x+1$$

Combine 15x and 4x to get 19x.

$$5x^{3}-17x^{2}+19x-5-2+x+1$$

Subtract 2 from -5 to get -7.

$$5x^{3}-17x^{2}+19x-7+x+1$$

Combine 19x and x to get 20x.

$$5x^{3}-17x^{2}+20x-7+1$$

Add -7 and 1 to get -6.

$$5x^{3}-17x^{2}+20x-6$$