If `2x^(2) + ax + b`, when divide by x - 3, leaves a remainder of 31 and `x^(2) + bx + a`, when divided by x-3, leaves a remainder of 24, then a+b equals

Options

A. 8
B. `7`
C. 23
D. `-23`

Answer

To solve the problem step by step, we will use the Remainder Theorem, which states that the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \).

### Step 1: Set up the equations based on the given information.

1. For the polynomial \( 2x^2 + ax + b \) when divided by \( x - 3 \), the remainder is 31. Thus, we can write:
\[ f(3) = 2(3^2) + a(3) + b = 31 \]
Simplifying this:
\[ 2(9) + 3a + b = 31 \]
\[ 18 + 3a + b = 31 \]
Rearranging gives us:
\[ 3a + b = 31 - 18 \]
\[ 3a + b = 13 \quad \text{(Equation 1)} \]

2. For the polynomial \( x^2 + bx + a \) when divided by \( x - 3 \), the remainder is 24. Thus, we can write:
\[ g(3) = (3^2) + b(3) + a = 24 \]
Simplifying this:
\[ 9 + 3b + a = 24 \]
Rearranging gives us:
\[ 3b + a = 24 - 9 \]
\[ 3b + a = 15 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.

We have two equations:
1. \( 3a + b = 13 \) (Equation 1)
2. \( 3b + a = 15 \) (Equation 2)

We can solve these equations simultaneously. Let's express \( b \) from Equation 1:
\[ b = 13 - 3a \]

Now substitute \( b \) into Equation 2:
\[ 3(13 - 3a) + a = 15 \]
Expanding this:
\[ 39 - 9a + a = 15 \]
Combining like terms:
\[ 39 - 8a = 15 \]
Rearranging gives:
\[ -8a = 15 - 39 \]
\[ -8a = -24 \]
Dividing by -8:
\[ a = 3 \]

### Step 3: Find \( b \).

Now substitute \( a = 3 \) back into Equation 1 to find \( b \):
\[ 3(3) + b = 13 \]
\[ 9 + b = 13 \]
\[ b = 13 - 9 \]
\[ b = 4 \]

### Step 4: Calculate \( a + b \).

Now we can find \( a + b \):
\[ a + b = 3 + 4 = 7 \]

### Final Answer:
Thus, \( a + b = 7 \).

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