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 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 \(x + a\):

1. 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:
\[
(x^2 + (a+b)x + ab) - (x^2 + ax) = (a+b)x - ax + ab = bx + ab.
\]

2. 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**.