Question

determine the quotient and remainder. \\((x^2 - 12x + 27) \div (x - 9)\\)

Explanation:

Step1: Factor the numerator

We factor the quadratic expression \(x^2 - 12x + 27\). We need two numbers that multiply to \(27\) and add up to \(-12\). The numbers are \(-9\) and \(-3\), so \(x^2 - 12x + 27=(x - 9)(x - 3)\).

Step2: Perform the division

Now we divide \(\frac{(x - 9)(x - 3)}{x - 9}\). Since \(x
eq9\) (to avoid division by zero), we can cancel out the common factor \((x - 9)\) from the numerator and the denominator. So the quotient is \(x - 3\) and the remainder is \(0\) (because the division results in a polynomial with no remainder).

Answer:

Quotient: \(x - 3\), Remainder: \(0\)