what polynomial identities would result from dividing x^2+ (a+b)x+ab by x+a. Enter the option number of the correct answer.

Question

option 1: x^2+(a+b)x+ab= (x+a)(x-b)
option 2: x^2+(a+b)x+ab= (x-a)(x-b)
option 3: x^2+(a+b)x+ab= (x+a)(x+b)

Answer 1

To divide \(x^2 + (a+b)x + ab\) by \(x + a\), we can use polynomial long division or synthetic division.

Starting with the polynomial:

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

When we divide it by \(x + a\), we look for a quotient polynomial \(Q(x)\) such that:

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

Using polynomial long division, we find:

1. The leading term \(Q(x)\) should start as \(x + b\) because when we multiply \(x + a\) by \(x + b\), the leading term gives us \(x^2\).

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

When we expand this, we indeed retrieve \(x^2 + (a + b)x + ab\).

Thus, the result of our division shows that:

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

Therefore, the correct option is:

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