1. If $$ \frac{x^{2}+13 x+15}{(x-1)\left(x^{2}+2\right)}=\frac{A}{x-1}+\frac{B x+C}{x^{2}+2} $$, then analyse and assess the descending order of $$ A, B, C $$.

Question

UpStudy AI · Accepted Answer

To analyze and determine the descending order of the constants $$ A $$, $$ B $$, and $$ C $$ in the partial fraction decomposition:

$$
\frac{x^{2} + 13x + 15}{(x-1)(x^{2}+2)} = \frac{A}{x-1} + \frac{Bx + C}{x^{2}+2}
$$

**Step 1: Multiply Both Sides by the Denominator**

Multiply both sides by $$(x-1)(x^{2}+2)$$ to eliminate the denominators:

$$
x^{2} + 13x + 15 = A(x^{2} + 2) + (Bx + C)(x - 1)
$$

**Step 2: Expand the Right-Hand Side**

Expand and simplify the right-hand side:

$$
A x^{2} + 2A + Bx(x) - Bx + Cx - C = (A + B)x^{2} + (-B + C)x + (2A - C)
$$

**Step 3: Set Up the System of Equations**

Equate the coefficients of corresponding powers of $$ x $$:

$$
\begin{cases}
A + B = 1 & \text{(Coefficient of } x^{2}) \
-B + C = 13 & \text{(Coefficient of } x) \
2A - C = 15 & \text{(Constant term)}
\end{cases}
$$

**Step 4: Solve the System**

1. From the first equation: $$ A = 1 - B $$.
2. Substitute $$ A = 1 - B $$ into the third equation:
   
   $$
   2(1 - B) - C = 15 \
   2 - 2B - C = 15 \
   -2B - C = 13
   $$

3. From the second equation: $$ C = B + 13 $$.

4. Substitute $$ C = B + 13 $$ into $$ -2B - C = 13 $$:

$$
   -2B - (B + 13) = 13 \
   -3B - 13 = 13 \
   -3B = 26 \
   B = -\frac{26}{3}
   $$

5. Now, find $$ A $$ and $$ C $$:

$$
   A = 1 - \left(-\frac{26}{3}\right) = \frac{29}{3}
   $$
   
   $$
   C = -\frac{26}{3} + 13 = \frac{13}{3}
   $$

**Step 5: Arrange in Descending Order**

Now, we have:

$$
A = \frac{29}{3} \approx 9.67 \
C = \frac{13}{3} \approx 4.33 \
B = -\frac{26}{3} \approx -8.67
$$

Arranging them from largest to smallest:

$$
A > C > B
$$

**Conclusion:**

The descending order of the constants is:

$$
A, \quad C, \quad B
$$

**Answer:**  
After simplifying, A is largest, then C, and then B. The descending order is A , C, B.
The descending order of the constants is A, C, B.

If , then analyse and assess the descending order of .

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions