We are asked to perform the partial fraction decomposition of the following rational expression:

$\frac{5x^2 - 24x + 6}{x(x+2)(x+3)}$

Step-by-Step Process

1. Set up the partial fraction decomposition:

Since the denominator is a product of three linear factors $x$, $x+2$, and $x+3$, the decomposition will take the form:

$\frac{5x^2 - 24x + 6}{x(x+2)(x+3)} = \frac{A}{x} + \frac{B}{x+2} + \frac{C}{x+3}$

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

2. Multiply both sides by the denominator $x(x+2)(x+3)$ to eliminate the denominators:

$5x^2 - 24x + 6 = A(x+2)(x+3) + Bx(x+3) + Cx(x+2)$

3. Expand the right-hand side:

Expand each term on the right-hand side:

• $A(x+2)(x+3) = A(x^2 + 5x + 6)$
• $B(x)(x+3) = B(x^2 + 3x)$
• $C(x)(x+2) = C(x^2 + 2x)$

Thus, the equation becomes:

$5x^2 - 24x + 6 = A(x^2 + 5x + 6) + B(x^2 + 3x) + C(x^2 + 2x)$

4. Combine like terms:

Collect all terms involving $x^2$, $x$, and constants on the right-hand side:

$5x^2 - 24x + 6 = (A + B + C)x^2 + (5A + 3B + 2C)x + 6A$

5. Set up a system of equations:

Now, equate the coefficients of like powers of $x$ on both sides:

• For $x^2$: $A + B + C = 5$
• For $x$: $5A + 3B + 2C = -24$
• For the constant: $6A = 6$

6. Solve the system of equations:

From the constant equation, $6A = 6$, we find:

$A = 1$

Substitute $A = 1$ into the other two equations:

• $1 + B + C = 5$ → $B + C = 4$
• $5(1) + 3B + 2C = -24$ → $5 + 3B + 2C = -24$ → $3B + 2C = -29$

Now we have the system:

• $B + C = 4$
• $3B + 2C = -29$

Solve this system using substitution or elimination:

From $B + C = 4$, we have $C = 4 - B$. Substitute this into the second equation:

$3B + 2(4 - B) = -29$ $3B + 8 - 2B = -29$ $B + 8 = -29$ $B = -37$

Now, substitute $B = -37$ into $B + C = 4$:

$-37 + C = 4$ $C = 41$

7. Write the partial fraction decomposition:

Now that we have $A = 1$, $B = -37$, and $C = 41$, the partial fraction decomposition is:

$\frac{5x^2 - 24x + 6}{x(x+2)(x+3)} = \frac{1}{x} - \frac{37}{x+2} + \frac{41}{x+3}$

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Would you like more details on this decomposition or have any other questions?

Here are five related questions for practice:

1. How do you perform partial fraction decomposition when there are repeated factors in the denominator?
2. How would you decompose a rational expression with an irreducible quadratic factor in the denominator?
3. Can you find the partial fraction decomposition of $\frac{x^2 - 5x + 4}{(x-1)(x+1)^2}$?
4. How does partial fraction decomposition help in integration?
5. What is the difference between proper and improper fractions in partial fraction decomposition?

Tip: Always check if the numerator's degree is less than the denominator's when starting partial fraction decomposition. If not, perform polynomial long division first.