Solve [1/a-1+1/a-2-1/(1-a)(a-2)]*[1/a-1-frac{1}{2-a}+frac{1}{(1-a)(a-2)}]*(a^2-3a+2)

$$[ \frac { 1 } { a - 1 } + \frac { 1 } { a - 2 } - \frac { 1 } { ( 1 - a ) ( a - 2 ) } ] \cdot [ \frac { 1 } { a - 1 } - \frac { 1 } { 2 - a } + \frac { 1 } { ( 1 - a ) ( a - 2 ) } ] \cdot ( a ^ { 2 } - 3 a + 2 )$$

Evaluate

$$4$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and a-2 is \left(a-2\right)\left(a-1\right). Multiply \frac{1}{a-1} times \frac{a-2}{a-2}. Multiply \frac{1}{a-2} times \frac{a-1}{a-1}.

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

Since \frac{a-2}{\left(a-2\right)\left(a-1\right)} and \frac{a-1}{\left(a-2\right)\left(a-1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in a-2+a-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)\left(a-1\right) and \left(1-a\right)\left(a-2\right) is \left(a-2\right)\left(a-1\right). Multiply \frac{1}{\left(1-a\right)\left(a-2\right)} times \frac{-1}{-1}.

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

Since \frac{2a-3}{\left(a-2\right)\left(a-1\right)} and \frac{-1}{\left(a-2\right)\left(a-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2a-3-\left(-1\right).

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

Combine like terms in 2a-3+1.

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

Factor the expressions that are not already factored in \frac{2a-2}{\left(a-2\right)\left(a-1\right)}.

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

Cancel out a-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and 2-a is \left(a-1\right)\left(-a+2\right). Multiply \frac{1}{a-1} times \frac{-a+2}{-a+2}. Multiply \frac{1}{2-a} times \frac{a-1}{a-1}.

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

Since \frac{-a+2}{\left(a-1\right)\left(-a+2\right)} and \frac{a-1}{\left(a-1\right)\left(-a+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -a+2-\left(a-1\right).

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

Combine like terms in -a+2-a+1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-1\right)\left(-a+2\right) and \left(1-a\right)\left(a-2\right) is \left(a-2\right)\left(a-1\right). Multiply \frac{-2a+3}{\left(a-1\right)\left(-a+2\right)} times \frac{-1}{-1}. Multiply \frac{1}{\left(1-a\right)\left(a-2\right)} times \frac{-1}{-1}.

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

Since \frac{-\left(-2a+3\right)}{\left(a-2\right)\left(a-1\right)} and \frac{-1}{\left(a-2\right)\left(a-1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -\left(-2a+3\right)-1.

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

Combine like terms in 2a-3-1.

$$\frac{2}{a-2}\times \frac{2a-4}{\left(a-2\right)\left(a-1\right)}\left(a^{2}-3a+2\right)$$

Factor the expressions that are not already factored in \frac{2a-4}{\left(a-2\right)\left(a-1\right)}.

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

Cancel out a-2 in both numerator and denominator.

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

Multiply \frac{2}{a-2} times \frac{2}{a-1} by multiplying numerator times numerator and denominator times denominator.

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

Express \frac{2\times 2}{\left(a-2\right)\left(a-1\right)}\left(a^{2}-3a+2\right) as a single fraction.

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

Multiply 2 and 2 to get 4.

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

Factor the expressions that are not already factored.

$$\frac{4\left(a-2\right)\left(a-1\right)}{\left(a-2\right)\left(a-1\right)}$$

Cancel out \left(a-2\right)\left(a-1\right) in both numerator and denominator.

$$4$$

Factor

$$2^{2}$$