To determine which option represents the polynomial identity that results from dividing \( x^2 + (a+b)x + ab \) by \( x + a \), we can perform polynomial long division or check the given options directly.
Let \( P(x) = x^2 + (a+b)x + ab \).
- Dividing \( P(x) \) by \( x + a \):
- The first term of \( P(x) \) is \( x^2 \). When we divide \( x^2 \) by \( x \), we get \( x \).
- Next, we multiply \( x + a \) by \( x \): \[ x(x + a) = x^2 + ax \]
- Now, we subtract this from \( P(x) \): \[ (x^2 + (a+b)x + ab) - (x^2 + ax) = (a+b)x - ax + ab = bx + ab \]
- Now we divide \( bx + ab \) by \( x + a \):
- The first term is \( bx \), which gives \( b \) when divided by \( x \).
- Multiplying \( x + a \) by \( b \): \[ b(x + a) = bx + ab \]
- Subtracting gives us a remainder of \( 0 \): \[ (bx + ab) - (bx + ab) = 0 \]
- Thus, we have the identity: \[ x^2 + (a+b)x + ab = (x + a)(x + b) \]
Now, let's verify which of the options correspond to this result:
- Option #1: \( x^2 + (a+b)x + ab = (x + a)(x - b) \) is incorrect.
- Option #2: \( x^2 + (a+b)x + ab = (x - a)(x + b) \) is incorrect.
- Option #3: \( x^2 + (a+b)x + ab = (x + a)(x + b) \) is correct.
Thus, the correct answer is Option #3.
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