To determine the correct polynomial identity, we will divide the polynomial \(x^2 + (a+b)x + ab\) by \(x + a\) and check which of the given options is a valid factorization.
Let's begin by performing synthetic or long division of \(x^2 + (a+b)x + ab\) by \(x + a\):
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First, set up the division:
- Divide the leading term: \(x^2 \div x = x\).
- Multiply \(x\) by \(x + a\): \(x \cdot (x + a) = x^2 + ax\).
- Subtract from the original polynomial: \[ (x^2 + (a+b)x + ab) - (x^2 + ax) = (a+b)x - ax + ab = bx + ab. \]
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Next, divide the new leading term \(bx\) by \(x + a\):
- Divide \(bx \div x = b\).
- Multiply \(b\) by \(x + a\): \(b \cdot (x + a) = bx + ab\).
- Subtract: \[ (bx + ab) - (bx + ab) = 0. \]
Thus, we have confirmed that the division yields a quotient of \(x + b\) with a remainder of \(0\). This means:
\[ x^2 + (a+b)x + ab = (x + a)(x + b). \]
Now let's match this with the options:
- Option #1: \(x^2 + (a+b)x + ab = (x + a)(x - b)\) - Incorrect.
- Option #2: \(x^2 + (a+b)x + ab = (x - a)(x + b)\) - Incorrect.
- Option #3: \(x^2 + (a+b)x + ab = (x + a)(x + b)\) - Correct.
Therefore, the correct option is Option #3.
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