To solve for the values of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] for the partial fraction decomposition of the given function, we follow these steps:
1. Given Expression: We start with the fraction:
[tex]\[
\frac{x}{x^2 - 5x + 6}
\][/tex]
2. Factor the Denominator: The denominator [tex]\( x^2 - 5x + 6 \)[/tex] factors into:
[tex]\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\][/tex]
3. Set up the Partial Fractions: We express the fraction as a sum of partial fractions with unknown coefficients [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
\frac{x}{(x-2)(x-3)} = \frac{A}{x-2} + \frac{B}{x-3}
\][/tex]
4. Combine the Fractions on the Right-Hand Side: To combine the fractions, we write:
[tex]\[
\frac{A}{x-2} + \frac{B}{x-3} = \frac{A(x-3) + B(x-2)}{(x-2)(x-3)}
\][/tex]
5. Set the Numerators Equal: Since the denominators are the same, we can equate the numerators:
[tex]\[
x = A(x-3) + B(x-2)
\][/tex]
6. Expand the Right-Hand Side: Distribute [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
x = Ax - 3A + Bx - 2B
\][/tex]
7. Combine Like Terms: Combine the terms involving [tex]\( x \)[/tex]:
[tex]\[
x = (A + B)x - 3A - 2B
\][/tex]
8. Set up a System of Equations: For the equation [tex]\( x = (A + B)x - 3A - 2B \)[/tex] to hold for all [tex]\( x \)[/tex], the coefficients on both sides must be equal. This gives us two simultaneous equations:
[tex]\[
A + B = 1
\][/tex]
[tex]\[
-3A - 2B = 0
\][/tex]
9. Solve the System of Equations:
- From the first equation, solve for [tex]\( A \)[/tex]:
[tex]\[
A = 1 - B
\][/tex]
- Substitute [tex]\( A = 1 - B \)[/tex] into the second equation:
[tex]\[
-3(1 - B) - 2B = 0
\][/tex]
- Simplify and solve for [tex]\( B \)[/tex]:
[tex]\[
-3 + 3B - 2B = 0
\][/tex]
[tex]\[
3B - 2B = 3
\][/tex]
[tex]\[
B = 3
\][/tex]
- Substitute [tex]\( B = 3 \)[/tex] back into [tex]\( A = 1 - B \)[/tex]:
[tex]\[
A = 1 - 3
\][/tex]
[tex]\[
A = -2
\][/tex]
Thus, the values of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are:
[tex]\[
A = -2 \quad \text{and} \quad B = 3
\][/tex]
Therefore, the correct answer is:
[tex]\[
(-2, 3)
\][/tex]
