Solve (a-1)^2(a-3)(a-2)-(aa)(a-1+a)

$$( a - 1 ) ^ { 2 } ( a - 3 ) ( a - 2 ) - ( a a ) ( a - 1 + a )$$

Evaluate

$$a^{4}-9a^{3}+18a^{2}-17a+6$$

Solution Steps

Multiply a and a to get a^{2}.

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

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-1\right)^{2}.

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

Use the distributive property to multiply a^{2}-2a+1 by a-3 and combine like terms.

$$\left(a^{3}-5a^{2}+7a-3\right)\left(a-2\right)-a^{2}\left(a-1+a\right)$$

Use the distributive property to multiply a^{3}-5a^{2}+7a-3 by a-2 and combine like terms.

$$a^{4}-7a^{3}+17a^{2}-17a+6-a^{2}\left(a-1+a\right)$$

Combine a and a to get 2a.

$$a^{4}-7a^{3}+17a^{2}-17a+6-a^{2}\left(2a-1\right)$$

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

$$a^{4}-7a^{3}+17a^{2}-17a+6-\left(2a^{3}-a^{2}\right)$$

To find the opposite of 2a^{3}-a^{2}, find the opposite of each term.

$$a^{4}-7a^{3}+17a^{2}-17a+6-2a^{3}+a^{2}$$

Combine -7a^{3} and -2a^{3} to get -9a^{3}.

$$a^{4}-9a^{3}+17a^{2}-17a+6+a^{2}$$

Combine 17a^{2} and a^{2} to get 18a^{2}.

$$a^{4}-9a^{3}+18a^{2}-17a+6$$

Expand

$$a^{4}-9a^{3}+18a^{2}-17a+6$$

Solution Steps

Multiply a and a to get a^{2}.

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

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-1\right)^{2}.

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

Use the distributive property to multiply a^{2}-2a+1 by a-3 and combine like terms.

$$\left(a^{3}-5a^{2}+7a-3\right)\left(a-2\right)-a^{2}\left(a-1+a\right)$$

Use the distributive property to multiply a^{3}-5a^{2}+7a-3 by a-2 and combine like terms.

$$a^{4}-7a^{3}+17a^{2}-17a+6-a^{2}\left(a-1+a\right)$$

Combine a and a to get 2a.

$$a^{4}-7a^{3}+17a^{2}-17a+6-a^{2}\left(2a-1\right)$$

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

$$a^{4}-7a^{3}+17a^{2}-17a+6-\left(2a^{3}-a^{2}\right)$$

To find the opposite of 2a^{3}-a^{2}, find the opposite of each term.

$$a^{4}-7a^{3}+17a^{2}-17a+6-2a^{3}+a^{2}$$

Combine -7a^{3} and -2a^{3} to get -9a^{3}.

$$a^{4}-9a^{3}+17a^{2}-17a+6+a^{2}$$

Combine 17a^{2} and a^{2} to get 18a^{2}.

$$a^{4}-9a^{3}+18a^{2}-17a+6$$