To resolve the given rational expression into partial fractions, we start with the expression:

$\frac{7x + 3}{(x-2)(x-6)}$

We can express this as a sum of two simpler fractions. Let's assume:

$\frac{7x + 3}{(x-2)(x-6)} = \frac{A}{x-2} + \frac{B}{x-6}$

where $A$ and $B$ are constants to be determined.

Step 1: Combine the Right-Hand Side into a Single Fraction

To find a common denominator on the right-hand side, we can write:

$\frac{A}{x-2} + \frac{B}{x-6} = \frac{A(x-6) + B(x-2)}{(x-2)(x-6)}$

Equating the numerators of both sides gives us:

$7x + 3 = A(x-6) + B(x-2)$

Step 2: Expand and Collect Like Terms

Expanding the right-hand side:

$A(x-6) + B(x-2) = Ax - 6A + Bx - 2B$

$= (A + B)x - (6A + 2B)$

Now we equate the coefficients of like terms from both sides of the equation:

$7x + 3 = (A + B)x - (6A + 2B)$

From this, we can set up two equations:

1. For the coefficient of $x$: $A + B = 7$

2. For the constant term: $-6A - 2B = 3$

Step 3: Solve the System of Equations

We have the system of equations:

1. $A + B = 7$
2. $-6A - 2B = 3$

From the first equation, we can express $B$ in terms of $A$:

$B = 7 - A$

Substitute $B$ into the second equation:

$-6A - 2(7 - A) = 3$

$-6A - 14 + 2A = 3$

$-4A - 14 = 3$

$-4A = 17$

$A = -\frac{17}{4}$

Now substitute $A$ back into the equation for $B$:

$B = 7 - \left(-\frac{17}{4}\right)$

$B = 7 + \frac{17}{4}$

$B = \frac{28}{4} + \frac{17}{4}$

$B = \frac{45}{4}$

Step 4: Write the Partial Fraction Decomposition

Using the values of $A$ and $B$ we found, the partial fraction decomposition is:

$\frac{7x + 3}{(x-2)(x-6)} = \frac{-\frac{17}{4}}{x-2} + \frac{\frac{45}{4}}{x-6}$

Simplifying:

$\frac{7x + 3}{(x-2)(x-6)} = -\frac{17}{4(x-2)} + \frac{45}{4(x-6)}$

Summary

The partial fraction decomposition of the given expression is:

$\frac{7x + 3}{(x-2)(x-6)} = -\frac{17}{4(x-2)} + \frac{45}{4(x-6)}$

Would you like more details, or do you have any questions about this process?

Relative Questions:

1. How can we verify the correctness of the partial fraction decomposition?
2. What is the significance of partial fraction decomposition in integration?
3. Can partial fractions be applied to non-linear factors in the denominator?
4. How would you handle repeated linear factors in the denominator?
5. What if the numerator's degree is higher than the denominator? How does that affect partial fraction decomposition?

Tip:

When solving partial fractions, always ensure that the degree of the numerator is less than the degree of the denominator before proceeding. If it's not, perform polynomial long division first.