What is the remainder when `13x^(2)+22x-10` is divided by `x+2`?

Options

A. 2
B. `-2`
C. 0
D. `-4`

Answer

To find the remainder when \( 13x^2 + 22x - 10 \) is divided by \( x + 2 \), we can use polynomial long division. Here’s a step-by-step solution:

### Step 1: Set up the division
We want to divide \( 13x^2 + 22x - 10 \) by \( x + 2 \).

### Step 2: Divide the leading term
Divide the leading term of the dividend \( 13x^2 \) by the leading term of the divisor \( x \):
\[ \frac{13x^2}{x} = 13x \]
This means \( 13x \) is the first term of our quotient.

### Step 3: Multiply and subtract
Now, multiply \( 13x \) by the entire divisor \( x + 2 \):
\[ 13x \cdot (x + 2) = 13x^2 + 26x \]
Next, subtract this from the original polynomial:
\[ (13x^2 + 22x - 10) - (13x^2 + 26x) = 22x - 26x - 10 = -4x - 10 \]

### Step 4: Repeat the process
Now, divide the leading term of the new polynomial \( -4x \) by the leading term of the divisor \( x \):
\[ \frac{-4x}{x} = -4 \]
So, \( -4 \) is the next term of our quotient.

### Step 5: Multiply and subtract again
Multiply \( -4 \) by the entire divisor \( x + 2 \):
\[ -4 \cdot (x + 2) = -4x - 8 \]
Now, subtract this from the current polynomial:
\[ (-4x - 10) - (-4x - 8) = -10 + 8 = -2 \]

### Step 6: Conclusion
Since the degree of the remainder \( -2 \) is less than the degree of the divisor \( x + 2 \), we stop here. The remainder when \( 13x^2 + 22x - 10 \) is divided by \( x + 2 \) is:
\[ \text{Remainder} = -2 \]

### Final Answer
The remainder is \( -2 \).

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