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) (1 point)

Answer 1

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 \( P(x) = x^2 + (a+b)x + ab \).

1. **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**.