To decompose the rational expression

$\frac{x^2 - 10x + 13}{(x - 1)(x - 5)(x - 2)},$

we will use partial fraction decomposition. The process involves expressing the fraction as a sum of simpler fractions, where each denominator is one of the linear factors in the original denominator. Here's how:

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Step 1: Write the general form for decomposition

The denominator is already factored as $(x - 1)(x - 5)(x - 2)$. The partial fraction decomposition will have the form:

$\frac{x^2 - 10x + 13}{(x - 1)(x - 5)(x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 5} + \frac{C}{x - 2},$

where $A$, $B$, and $C$ are constants we need to find.

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Step 2: Combine the terms into a single fraction

Multiply through by the denominator $(x - 1)(x - 5)(x - 2)$ to eliminate the fractions:

$x^2 - 10x + 13 = A(x - 5)(x - 2) + B(x - 1)(x - 2) + C(x - 1)(x - 5).$

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Step 3: Expand and simplify the right-hand side

Expand each term:

1. $A(x - 5)(x - 2) = A(x^2 - 7x + 10)$,
2. $B(x - 1)(x - 2) = B(x^2 - 3x + 2)$,
3. $C(x - 1)(x - 5) = C(x^2 - 6x + 5)$.

Thus, the equation becomes:

$x^2 - 10x + 13 = A(x^2 - 7x + 10) + B(x^2 - 3x + 2) + C(x^2 - 6x + 5).$

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Step 4: Group and equate coefficients

Combine like terms on the right-hand side:

$x^2 - 10x + 13 = (A + B + C)x^2 + (-7A - 3B - 6C)x + (10A + 2B + 5C).$

Equate the coefficients of $x^2$, $x$, and the constant term on both sides:

1. Coefficient of $x^2$: $A + B + C = 1$,
2. Coefficient of $x$: $-7A - 3B - 6C = -10$,
3. Constant term: $10A + 2B + 5C = 13$.

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Step 5: Solve the system of equations

We now solve the system of linear equations:

1. $A + B + C = 1,$
2. $-7A - 3B - 6C = -10,$
3. $10A + 2B + 5C = 13.$

Using substitution or matrix methods, solve this system to find $A$, $B$, and $C$. After solving, we get:

$A = 1, \quad B = -2, \quad C = 2.$

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Step 6: Write the final decomposition

The partial fraction decomposition is:

$\frac{x^2 - 10x + 13}{(x - 1)(x - 5)(x - 2)} = \frac{1}{x - 1} - \frac{2}{x - 5} + \frac{2}{x - 2}.$

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Would you like further details about any step?

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5 Related Questions

1. How do you solve systems of equations using matrices?
2. Can partial fraction decomposition be applied to improper fractions?
3. What if one of the factors in the denominator is repeated?
4. How do you integrate a decomposed fraction?
5. Why is partial fraction decomposition useful in calculus?

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Tip

To check your decomposition, combine the fractions back together and verify that you get the original expression!