Which of the following polynomial identities would result from dividing x2+(a+b)x+ab

Question

by x+a
? Enter the option number of the correct answer.

Option #1: x2+(a+b)x+ab=(x+a)(x−b)


Option #2: x2+(a+b)x+ab=(x−a)(x+b)


Option #3: x2+(a+b)x+ab=(x+a)(x+b)


(1 point)

Answer 1

To determine which option correctly represents the division of the polynomial \( x^2 + (a+b)x + ab \) by \( x + a \), we can perform polynomial long division or synthetic division.

Let \( P(x) = x^2 + (a+b)x + ab \) and divide it by \( x + a \).

1. Divide the leading term \( x^2 \) by \( x \) to get \( x \).
2. Multiply \( x + a \) by \( x \) resulting in \( x^2 + ax \).
3. Subtract \( (x^2 + ax) \) from \( P(x) \):

\[ (x^2 + (a+b)x + ab) - (x^2 + ax) = (a+b)x - ax + ab = bx + ab \]

4. Next, divide the leading term \( bx \) by \( x \) to get \( b \).
5. Multiply \( x + a \) by \( b \) resulting in \( bx + ab \).
6. Subtract \( (bx + ab) \) from the previous result:

\[ (bx + ab) - (bx + ab) = 0 \]

Since we have reached 0, the division is exact and we can express it as:

\[ x^2 + (a+b)x + ab = (x + a)(x + b) \]

Thus, the correct polynomial identity is:

\[ x^2 + (a+b)x + ab = (x + a)(x + b) \]

This corresponds to Option #3.

So, the answer is: 3.