The polynomial `px^(3)+4x^(2)-3x+q` completely divisible by `x^2-1:` find the values of p and q. Also for these values of p and q factorize the given polynomial completely.

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) such that the polynomial \( f(x) = px^3 + 4x^2 - 3x + q \) is completely divisible by \( x^2 - 1 \).

### Step 1: Understand the Divisibility Condition
Since \( f(x) \) is divisible by \( x^2 - 1 \), it means that \( f(x) \) has roots at \( x = 1 \) and \( x = -1 \). Therefore, \( f(1) = 0 \) and \( f(-1) = 0 \).

### Step 2: Calculate \( f(-1) \)
Substituting \( x = -1 \) into the polynomial:
\[ f(-1) = p(-1)^3 + 4(-1)^2 - 3(-1) + q \]
This simplifies to:
\[ f(-1) = -p + 4 + 3 + q = -p + q + 7 \]
Since \( f(-1) = 0 \):
\[ -p + q + 7 = 0 \quad \text{(Equation 1)} \]

### Step 3: Calculate \( f(1) \)
Now substituting \( x = 1 \) into the polynomial:
\[ f(1) = p(1)^3 + 4(1)^2 - 3(1) + q \]
This simplifies to:
\[ f(1) = p + 4 - 3 + q = p + q + 1 \]
Since \( f(1) = 0 \):
\[ p + q + 1 = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( -p + q + 7 = 0 \)
2. \( p + q + 1 = 0 \)

From Equation 2, we can express \( q \) in terms of \( p \):
\[ q = -p - 1 \quad \text{(Substituting into Equation 1)} \]

Substituting \( q \) into Equation 1:
\[ -p + (-p - 1) + 7 = 0 \]
This simplifies to:
\[ -2p + 6 = 0 \]
Solving for \( p \):
\[ 2p = 6 \implies p = 3 \]

### Step 5: Find \( q \)
Now substituting \( p = 3 \) back into the expression for \( q \):
\[ q = -3 - 1 = -4 \]

### Step 6: Final Values
Thus, we have:
\[ p = 3, \quad q = -4 \]

### Step 7: Factor the Polynomial
Now substituting \( p \) and \( q \) back into the polynomial:
\[ f(x) = 3x^3 + 4x^2 - 3x - 4 \]

To factor \( f(x) \), we can use polynomial long division with the divisor \( x^2 - 1 \):
1. Divide \( 3x^3 \) by \( x^2 \) to get \( 3x \).
2. Multiply \( 3x \) by \( x^2 - 1 \) to get \( 3x^3 - 3x \).
3. Subtract this from \( f(x) \):
\[ (3x^3 + 4x^2 - 3x - 4) - (3x^3 - 3x) = 4x^2 - 4 \]
4. Now divide \( 4x^2 \) by \( x^2 \) to get \( 4 \).
5. Multiply \( 4 \) by \( x^2 - 1 \) to get \( 4x^2 - 4 \).
6. Subtract this from the previous result:
\[ (4x^2 - 4) - (4x^2 - 4) = 0 \]

Thus, we have:
\[ f(x) = (x^2 - 1)(3x + 4) \]

### Final Factorization
The complete factorization of the polynomial is:
\[ f(x) = (x + 1)(x - 1)(3x + 4) \]