If x= 2 is a zero of polynomial `ax^2 - bx + 2` then what is the relation between a and b?

Options

A. 2a-b+ 1 = 0
B. a + b + 1 = 0
C. a-b+1 = 0
D. 7a-5b + 1 = 0

Answer

To find the relation between \( a \) and \( b \) given that \( x = 2 \) is a zero of the polynomial \( ax^2 - bx + 2 \), we can follow these steps:

### Step 1: Substitute \( x = 2 \) into the polynomial
Since \( x = 2 \) is a zero of the polynomial, substituting \( x = 2 \) into the polynomial should yield zero:
\[ p(2) = a(2^2) - b(2) + 2 = 0 \]

### Step 2: Simplify the equation
Calculating \( 2^2 \):
\[ p(2) = a(4) - 2b + 2 = 0 \]
This simplifies to:
\[ 4a - 2b + 2 = 0 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate terms involving \( a \) and \( b \):
\[ 4a - 2b = -2 \]

### Step 4: Divide the entire equation by 2
To simplify further, we can divide the entire equation by 2:
\[ 2a - b = -1 \]

### Step 5: Express the relationship
From this equation, we can express the relationship between \( a \) and \( b \):
\[ b = 2a + 1 \]

Thus, the relation between \( a \) and \( b \) is:
\[ b = 2a + 1 \]