If `(x^(4))/((x-a)(x-b)(x-c))= P(x)+(A)/(x-a)+(B)/(x-b)+(C)/(x-c)` then `P(x)=`

Options

A. `x-a`
B. `x-a-b`
C. `x-a-b-c`
D. `x+a+b+c`

Answer

To solve the equation

\[ \frac{x^4}{(x-a)(x-b)(x-c)} = P(x) + \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c} \]

we need to find \( P(x) \).

### Step 1: Identify the form of \( P(x) \)

Since the degree of the numerator \( x^4 \) is higher than the degree of the denominator \( (x-a)(x-b)(x-c) \) which is a cubic polynomial, we can express \( P(x) \) as a linear polynomial:

\[ P(x) = mx + n \]

### Step 2: Rewrite the equation

Substituting \( P(x) \) into the original equation gives:

\[ \frac{x^4}{(x-a)(x-b)(x-c)} = mx + n + \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c} \]

### Step 3: Combine the right-hand side

To combine the right-hand side, we need a common denominator:

\[ mx + n + \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c} = \frac{(mx+n)(x-a)(x-b)(x-c) + A(x-b)(x-c) + B(x-a)(x-c) + C(x-a)(x-b)}{(x-a)(x-b)(x-c)} \]

### Step 4: Set the numerators equal

Now, we have:

\[ x^4 = (mx+n)(x-a)(x-b)(x-c) + A(x-b)(x-c) + B(x-a)(x-c) + C(x-a)(x-b) \]

### Step 5: Expand the right-hand side

We need to expand the right-hand side and compare coefficients. The left-hand side is \( x^4 \), and we will expand the right-hand side to find coefficients of \( x^4, x^3, x^2, x, \) and the constant term.

### Step 6: Coefficient comparison

1. The coefficient of \( x^4 \) on the left is \( 1 \).
2. The coefficient of \( x^4 \) on the right is \( m \) (from \( (mx+n)(x-a)(x-b)(x-c) \)).
3. Therefore, we have:

\[ m = 1 \]

### Step 7: Find \( n \)

Next, we compare the coefficients of \( x^3 \):

1. The coefficient of \( x^3 \) on the left is \( 0 \).
2. The coefficient of \( x^3 \) on the right is \( -a - b - c + n \).
3. Setting these equal gives:

\[ -a - b - c + n = 0 \implies n = a + b + c \]

### Step 8: Write the final expression for \( P(x) \)

Now we can substitute back the values of \( m \) and \( n \) into \( P(x) \):

\[ P(x) = mx + n = 1 \cdot x + (a + b + c) = x + (a + b + c) \]

### Final Answer

Thus, we have:

\[ P(x) = x + a + b + c \]

---