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 answer

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.