Divide the polynomial x^3 - 6x^2 + 11x - 6 by (x - 2). What is the quotient?

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Multiple Choice

Divide the polynomial x^3 - 6x^2 + 11x - 6 by (x - 2). What is the quotient?

Explanation:
Dividing by a binomial like x − 2 reduces the degree by one, so you expect a quadratic as the quotient. A quick method is synthetic division with the value 2. Bring down the leading coefficient 1. Then multiply by 2 and add successively: 1 stays, 1×2 = 2 added to −6 gives −4, −4×2 = −8 added to 11 gives 3, 3×2 = 6 added to −6 gives 0 remainder. The resulting coefficients are 1, −4, 3, so the quotient is x^2 − 4x + 3. You can also see this by factoring the cubic as (x−1)(x−2)(x−3); removing the (x−2) factor leaves (x−1)(x−3) = x^2 − 4x + 3, which matches the quotient.

Dividing by a binomial like x − 2 reduces the degree by one, so you expect a quadratic as the quotient. A quick method is synthetic division with the value 2. Bring down the leading coefficient 1. Then multiply by 2 and add successively: 1 stays, 1×2 = 2 added to −6 gives −4, −4×2 = −8 added to 11 gives 3, 3×2 = 6 added to −6 gives 0 remainder. The resulting coefficients are 1, −4, 3, so the quotient is x^2 − 4x + 3. You can also see this by factoring the cubic as (x−1)(x−2)(x−3); removing the (x−2) factor leaves (x−1)(x−3) = x^2 − 4x + 3, which matches the quotient.

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