# How do you write the partial fraction decomposition of the rational expression # (x) / (x^(3)-x^(2)-2x +2)#?

Factor the bottom:

We have:

Combine the first two equations.

Add to the third equation.

Therefore

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To write the partial fraction decomposition of the rational expression ( \frac{x}{x^3 - x^2 - 2x + 2} ), we first factor the denominator.

The denominator can be factored as ( (x - 1)(x^2 - 2) ), which further simplifies to ( (x - 1)(x + \sqrt{2})(x - \sqrt{2}) ) using the difference of squares formula.

Now, we express the rational expression as a sum of simpler fractions with these as denominators. The general form of the partial fraction decomposition is:

[ \frac{x}{(x - 1)(x + \sqrt{2})(x - \sqrt{2})} = \frac{A}{x - 1} + \frac{B}{x + \sqrt{2}} + \frac{C}{x - \sqrt{2}} ]

To find the values of ( A ), ( B ), and ( C ), we multiply both sides by the denominator to clear the fractions:

[ x = A(x + \sqrt{2})(x - \sqrt{2}) + B(x - 1)(x - \sqrt{2}) + C(x - 1)(x + \sqrt{2}) ]

Next, we can solve for ( A ), ( B ), and ( C ) by substituting specific values for ( x ) that make the other terms drop out. For example, setting ( x = 1 ) eliminates the terms with ( B ) and ( C ), allowing us to solve for ( A ). Similarly, setting ( x = -\sqrt{2} ) and ( x = \sqrt{2} ) will help find ( B ) and ( C ), respectively.

After solving for ( A ), ( B ), and ( C ), we substitute these values back into the decomposition to get the final partial fraction decomposition of ( \frac{x}{x^3 - x^2 - 2x + 2} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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