Question

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

Answer 1

To determine which polynomial identity results from dividing \( x^2 + (a+b)x + ab \) by \( x+a \), we can use polynomial long division.

We start with the polynomial:

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

We want to divide this by:

\[
x + a
\]

1. The first term in the division would be \( x \) (as \( x^2 / x = x \)).
2. Multiply \( x \) by \( x + a \):

\[
x \cdot (x + a) = x^2 + ax
\]

3. Subtract this result from the original polynomial:

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

4. Now divide \( bx + ab \) by \( x + a \). The leading term \( bx/x = b \), so we add \( b \) to our quotient.
5. Multiply \( b \) by \( x + a \):

\[
b \cdot (x + a) = bx + ab
\]

6. Subtract this from \( bx + ab \):

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

The result of the division has no remainder, indicating that:

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

Thus, the correct option is:

**Option #3: \( x^2 + (a+b)x + ab = (x+a)(x+b) \)**.