Value of a and `b(a gt 0, b lt 0)` for which `8x^(3)-ax^(2)+54x+b` is a perfect cube is

Options

A. a=12, b=-9
B. a=36, b=-27
C. a=18, b=-27
D. a=16, b=-9

Answer

To find the values of \( a \) and \( b \) such that the polynomial \( 8x^3 - ax^2 + 54x + b \) is a perfect cube, we can follow these steps:

### Step 1: Understand the Structure of a Perfect Cube
A polynomial is a perfect cube if it can be expressed in the form \( (px + q)^3 \). Expanding this gives:
\[ (px + q)^3 = p^3x^3 + 3p^2qx^2 + 3pq^2x + q^3 \]

### Step 2: Match Coefficients
From the expansion, we can see that:
- The coefficient of \( x^3 \) is \( p^3 \).
- The coefficient of \( x^2 \) is \( 3p^2q \).
- The coefficient of \( x \) is \( 3pq^2 \).
- The constant term is \( q^3 \).

Given the polynomial \( 8x^3 - ax^2 + 54x + b \), we can identify:
- \( p^3 = 8 \)
- \( 3p^2q = -a \)
- \( 3pq^2 = 54 \)
- \( q^3 = b \)

### Step 3: Solve for \( p \)
From \( p^3 = 8 \), we find:
\[ p = 2 \]

### Step 4: Substitute \( p \) to Find \( q \)
Now, substituting \( p = 2 \) into the equation for \( 3pq^2 = 54 \):
\[ 3(2)q^2 = 54 \implies 6q^2 = 54 \implies q^2 = 9 \implies q = 3 \text{ or } q = -3 \]

### Step 5: Calculate \( a \) and \( b \)
Now we can find \( a \) using \( 3p^2q = -a \):
1. If \( q = 3 \):
\[ 3(2^2)(3) = -a \implies 3(4)(3) = -a \implies 36 = -a \implies a = -36 \text{ (not valid since } a > 0\text{)} \]
2. If \( q = -3 \):
\[ 3(2^2)(-3) = -a \implies 3(4)(-3) = -a \implies -36 = -a \implies a = 36 \]

Now, calculate \( b \):
\[ q^3 = b \implies (-3)^3 = b \implies b = -27 \]

### Step 6: Final Values
Thus, we have:
- \( a = 36 \) (which is greater than 0)
- \( b = -27 \) (which is less than 0)

### Conclusion
The values of \( a \) and \( b \) are:
\[ \boxed{a = 36, b = -27} \]