If `2x^(3) + ax^(2) + bx -2` leaves the remainders 7 and 0 when divided by (2x — 3) and (x + 2), respectively, then the values of a and b are respectively:

Options

A. `3, -3`
B. `-2,2`
C. `-3,3`
D. `2,-2`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given the polynomial \( 2x^3 + ax^2 + bx - 2 \) leaves a remainder of 7 when divided by \( 2x - 3 \) and a remainder of 0 when divided by \( x + 2 \).

### Step 1: Use the Remainder Theorem
According to the Remainder Theorem, if a polynomial \( P(x) \) is divided by \( x - c \), the remainder is \( P(c) \).

### Step 2: Find \( x \) for \( 2x - 3 = 0 \)
Set \( 2x - 3 = 0 \) to find \( x \):
\[ 2x = 3 \implies x = \frac{3}{2} \]

### Step 3: Substitute \( x = \frac{3}{2} \) into the polynomial
Substituting \( x = \frac{3}{2} \) into the polynomial:
\[ P\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right)^3 + a\left(\frac{3}{2}\right)^2 + b\left(\frac{3}{2}\right) - 2 \]
Calculating each term:
\[ = 2 \cdot \frac{27}{8} + a \cdot \frac{9}{4} + b \cdot \frac{3}{2} - 2 \]
\[ = \frac{27}{4} + \frac{9a}{4} + \frac{3b}{2} - 2 \]
Setting this equal to 7 (the remainder):
\[ \frac{27}{4} + \frac{9a}{4} + \frac{3b}{2} - 2 = 7 \]

### Step 4: Simplify the equation
Convert -2 to a fraction:
\[ -2 = -\frac{8}{4} \]
Thus, the equation becomes:
\[ \frac{27}{4} + \frac{9a}{4} + \frac{3b}{2} - \frac{8}{4} = 7 \]
Combine the fractions:
\[ \frac{19 + 9a + 6b}{4} = 7 \]
Multiply through by 4:
\[ 19 + 9a + 6b = 28 \]
Rearranging gives:
\[ 9a + 6b = 9 \quad \text{(Equation 1)} \]

### Step 5: Find \( x \) for \( x + 2 = 0 \)
Set \( x + 2 = 0 \) to find \( x \):
\[ x = -2 \]

### Step 6: Substitute \( x = -2 \) into the polynomial
Substituting \( x = -2 \):
\[ P(-2) = 2(-2)^3 + a(-2)^2 + b(-2) - 2 \]
Calculating each term:
\[ = 2(-8) + a(4) - 2b - 2 \]
\[ = -16 + 4a - 2b - 2 \]
Setting this equal to 0 (the remainder):
\[ -16 + 4a - 2b - 2 = 0 \]

### Step 7: Simplify the equation
Combine the constants:
\[ 4a - 2b - 18 = 0 \]
Rearranging gives:
\[ 4a - 2b = 18 \quad \text{(Equation 2)} \]

### Step 8: Solve the system of equations
We have the two equations:
1. \( 9a + 6b = 9 \)
2. \( 4a - 2b = 18 \)

From Equation 2, we can express \( b \) in terms of \( a \):
\[ 2b = 4a - 18 \implies b = 2a - 9 \]

### Step 9: Substitute \( b \) into Equation 1
Substituting \( b \) into Equation 1:
\[ 9a + 6(2a - 9) = 9 \]
Expanding:
\[ 9a + 12a - 54 = 9 \]
Combining like terms:
\[ 21a - 54 = 9 \]
Adding 54 to both sides:
\[ 21a = 63 \implies a = 3 \]

### Step 10: Find \( b \)
Substituting \( a = 3 \) back into the expression for \( b \):
\[ b = 2(3) - 9 = 6 - 9 = -3 \]

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ \boxed{3} \text{ and } \boxed{-3} \]