Hi, please could you show me the steps to simplify 5 /  ((x-2)(x+3)(x+4))  via partial fractions, thanks.

Sure! To simplify the given expression \(\frac{5}{(x-2)(x+3)(x+4)}\) using partial fractions, we'll follow these steps:

### Step 1: Set up the partial fraction form
Since the denominator \((x-2)(x+3)(x+4)\) consists of three distinct linear factors, we can express the fraction as a sum of partial fractions in the following form:

\[
\frac{5}{(x-2)(x+3)(x+4)} = \frac{A}{x-2} + \frac{B}{x+3} + \frac{C}{x+4}
\]

Here, \(A\), \(B\), and \(C\) are constants that we need to find.

### Step 2: Multiply both sides by the denominator
Multiply both sides of the equation by the common denominator \((x-2)(x+3)(x+4)\) to eliminate the denominators on the right-hand side:

\[
5 = A(x+3)(x+4) + B(x-2)(x+4) + C(x-2)(x+3)
\]

### Step 3: Expand the right-hand side
Now, expand each term on the right-hand side:

- \(A(x+3)(x+4) = A(x^2 + 7x + 12)\)
- \(B(x-2)(x+4) = B(x^2 + 2x - 8)\)
- \(C(x-2)(x+3) = C(x^2 + x - 6)\)

Thus, we get:

\[
5 = A(x^2 + 7x + 12) + B(x^2 + 2x - 8) + C(x^2 + x - 6)
\]

### Step 4: Combine like terms
Now, group the terms according to powers of \(x\):

\[
5 = (A + B + C)x^2 + (7A + 2B + C)x + (12A - 8B - 6C)
\]

### Step 5: Set up a system of equations
Since the left-hand side of the equation is just 5 (which is a constant), we compare coefficients of \(x^2\), \(x\), and the constant terms on both sides of the equation:

- Coefficient of \(x^2\): \(A + B + C = 0\)
- Coefficient of \(x\): \(7A + 2B + C = 0\)
- Constant term: \(12A - 8B - 6C = 5\)

### Step 6: Solve the system of equations
We now solve this system of three equations:

1. \(A + B + C = 0\)
2. \(7A + 2B + C = 0\)
3. \(12A - 8B - 6C = 5\)

#### Substitution or elimination method
From equation (1), solve for \(C\):

\[
C = -A - B
\]

Substitute \(C = -A - B\) into equations (2) and (3):

**For equation (2):**

\[
7A + 2B + (-A - B) = 0
\]

Simplify:

\[
6A + B = 0 \quad \Rightarrow \quad B = -6A
\]

**For equation (3):**

\[
12A - 8(-6A) - 6(-A - B) = 5
\]

Simplify:

\[
12A + 48A + 6A + 6B = 5
\]

Since \(B = -6A\), we substitute this into the equation:

\[
66A = 5 \quad \Rightarrow \quad A = \frac{5}{66}
\]

Now that we have \(A\), substitute it into \(B

Hi, please could you show me the steps to simplify 5 / ((x-2)(x+3)(x+4)) via partial fractions, thanks.

Step-by-step guide to simplifying the expression 5 / ((x-2)(x+3)(x+4)) using partial fractions. This tutorial covers the decomposition process, solving systems of equations, and finding constants. Ideal for students in Grades 10-12 studying algebra and rational expressions.

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