Resolve into Partial Fractions
(v) `(2x^(3)+3)/(x^(2)-5x+6)`

Answer

To resolve the expression \(\frac{2x^3 + 3}{x^2 - 5x + 6}\) into partial fractions, we will follow these steps:

### Step 1: Factor the Denominator
First, we need to factor the denominator \(x^2 - 5x + 6\).

The expression can be factored as:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

### Step 2: Set Up the Partial Fraction Decomposition
Since the degree of the numerator (3) is greater than the degree of the denominator (2), we will first perform polynomial long division.

Performing long division of \(2x^3 + 3\) by \(x^2 - 5x + 6\):

1. Divide the leading term: \(\frac{2x^3}{x^2} = 2x\).
2. Multiply \(2x\) by the entire divisor: \(2x(x^2 - 5x + 6) = 2x^3 - 10x^2 + 12x\).
3. Subtract this from the original polynomial:
\[ (2x^3 + 0x^2 + 3) - (2x^3 - 10x^2 + 12x) = 10x^2 - 12x + 3 \]

Now we have:
\[ \frac{2x^3 + 3}{x^2 - 5x + 6} = 2x + \frac{10x^2 - 12x + 3}{(x - 2)(x - 3)} \]

### Step 3: Set Up the Partial Fractions for the Remainder
Now, we need to resolve the fraction \(\frac{10x^2 - 12x + 3}{(x - 2)(x - 3)}\) into partial fractions:
\[ \frac{10x^2 - 12x + 3}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3} \]

### Step 4: Clear the Denominator
Multiply through by the denominator \((x - 2)(x - 3)\):
\[ 10x^2 - 12x + 3 = A(x - 3) + B(x - 2) \]

### Step 5: Expand and Collect Like Terms
Expanding the right side:
\[ A(x - 3) + B(x - 2) = Ax - 3A + Bx - 2B = (A + B)x + (-3A - 2B) \]

### Step 6: Set Up the System of Equations
Now, we can set up the equations by comparing coefficients:
1. \(A + B = 10\) (coefficient of \(x\))
2. \(-3A - 2B = -12\) (constant term)

### Step 7: Solve the System of Equations
From the first equation, we have:
\[ B = 10 - A \]

Substituting \(B\) into the second equation:
\[ -3A - 2(10 - A) = -12 \]
\[ -3A - 20 + 2A = -12 \]
\[ -A - 20 = -12 \]
\[ -A = 8 \implies A = -8 \]

Now substituting \(A\) back to find \(B\):
\[ B = 10 - (-8) = 18 \]

### Step 8: Write the Partial Fraction Decomposition
Now we can write the partial fraction decomposition:
\[ \frac{10x^2 - 12x + 3}{(x - 2)(x - 3)} = \frac{-8}{x - 2} + \frac{18}{x - 3} \]

### Step 9: Combine Everything
Putting it all together, we have:
\[ \frac{2x^3 + 3}{x^2 - 5x + 6} = 2x + \frac{-8}{x - 2} + \frac{18}{x - 3} \]

### Final Answer
Thus, the final answer is:
\[ \frac{2x^3 + 3}{x^2 - 5x + 6} = 2x - \frac{8}{x - 2} + \frac{18}{x - 3} \]