If the expression `px^3 - 2x^2 - qx + 18` is completely divisible by `(x^2- 9)`, what are the ratio between p and q respectively?

Options

A. `1:9`
B. `1:3`
C. `3:1`
D. `9:1`

Answer

To solve the problem, we need to determine the ratio between \( p \) and \( q \) given that the expression \( px^3 - 2x^2 - qx + 18 \) is completely divisible by \( x^2 - 9 \).

### Step 1: Understand the Divisibility Condition
Since \( x^2 - 9 \) is a factor of the polynomial, it means that the polynomial must equal zero for the roots of \( x^2 - 9 \). The roots are \( x = 3 \) and \( x = -3 \).

### Step 2: Substitute \( x = 3 \)
Substituting \( x = 3 \) into the polynomial:
\[ p(3)^3 - 2(3)^2 - q(3) + 18 = 0 \]
Calculating this gives:
\[ 27p - 18 - 3q + 18 = 0 \]
Simplifying:
\[ 27p - 3q = 0 \]
This can be rearranged to:
\[ 27p = 3q \]
Dividing both sides by 3:
\[ 9p = q \quad \text{(Equation 1)} \]

### Step 3: Substitute \( x = -3 \)
Now, substituting \( x = -3 \) into the polynomial:
\[ p(-3)^3 - 2(-3)^2 - q(-3) + 18 = 0 \]
Calculating this gives:
\[ -27p - 18 + 3q + 18 = 0 \]
Simplifying:
\[ -27p + 3q = 0 \]
This can be rearranged to:
\[ 27p = 3q \]
Dividing both sides by 3:
\[ 9p = q \quad \text{(Equation 2)} \]

### Step 4: Determine the Ratio
From both equations (Equation 1 and Equation 2), we have:
\[ q = 9p \]
To find the ratio of \( p \) to \( q \):
\[ \frac{p}{q} = \frac{p}{9p} = \frac{1}{9} \]

### Conclusion
Thus, the ratio of \( p \) to \( q \) is:
\[ \frac{p}{q} = \frac{1}{9} \]