Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
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]
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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
