Question

Simplify. $\frac{x^{3}-3 x^{2}+2 x-6}{x-3}$

   Simplify.
$\frac{x^{3}-3 x^{2}+2 x-6}{x-3}$

Algebra 2

Holliday, Luchin,… 1st Edition

Chapter 6, Problem 61

Instant Answer

Solved by Expert Heather Zimmers

03/16/2020

Step 1

We can group the first two terms and the last two terms separately. For the first two terms, we can factor out an $x^{2}$, and for the last two terms, we can factor out a 2. This gives us: \[x^{3}-3 x^{2}+2 x-6 = x^{2}(x-3) + 2(x-3)\] Show more…

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Simplify. $\frac{x^{3}-3 x^{2}+2 x-6}{x-3}$

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Key Concepts

Polynomial Division

This is a process by which a polynomial (the dividend) is divided by another polynomial (the divisor), analogous to integer division. In the context of simplifying rational expressions, polynomial division is used to express the rational expression as a polynomial plus a remainder divided by the divisor, and if the remainder is zero, it indicates that the divisor is a factor of the dividend.

Factor Theorem

This theorem states that for a polynomial f(x), if f(c) = 0 for some number c, then (x - c) is a factor of the polynomial. It is a useful concept for determining whether a particular expression can be factored out from a polynomial, which aids in the simplification of rational expressions.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form (x - c). It streamlines the division process and makes it easier to determine the quotient and remainder, which is especially beneficial when applying the factor theorem to check for factors and simplify rational expressions.

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