---
title: "Non-Repeated Linear Factors in College Algebra"
description: "Non-repeated linear factors are distinct denominator factors like (x-a) that appear once and give simple A/(x-a) terms in partial fraction decomposition."
canonical: "https://fiveable.me/college-algebra/key-terms/non-repeated-linear-factors"
type: "key-term"
subject: "College Algebra"
unit: "Unit 11"
---

# Non-Repeated Linear Factors in College Algebra

## Definition

Non-repeated linear factors are linear denominator factors that show up only once, like x-a. In College Algebra partial fraction decomposition, each one gets a term of the form A/(x-a).

## What It Is

Non-repeated linear factors are the distinct first-degree factors in a rational expression’s denominator that appear only one time. In College Algebra, you usually meet them when you break a rational expression into partial fractions.

If the denominator factors into pieces like (x-2)(x+5)(x-1), and none of those factors repeats, each factor is non-repeated. That means each one gets its own simple fraction in the decomposition: A/(x-2), B/(x+5), and C/(x-1). The unknown constants A, B, and C are what you solve for.

The reason this works is that a linear factor of degree 1 does not need a complicated numerator. For a non-repeated linear factor, the numerator is just a constant. You are matching the original rational expression by rewriting it as a sum of simpler pieces, then clearing denominators and comparing coefficients or plugging in values.

A common move is to set up an identity after multiplying by the least common denominator. For example, if
(2x+3)/[(x-1)(x+4)] = A/(x-1) + B/(x+4),
you multiply by (x-1)(x+4) and get 2x+3 = A(x+4) + B(x-1). From there, you solve for A and B.

The big thing to watch is repetition. If a factor appears more than once, the setup changes and you need extra terms like A/(x-a) + B/(x-a)^2. But if the factor is non-repeated, you only write one term for that factor, which keeps the algebra cleaner.

## Why It Matters

Non-repeated linear factors are the easiest part of partial fraction decomposition, so they set the pattern for the whole method. Once you can identify them, you can build the right decomposition without guessing extra powers or extra numerator variables.

That matters in College Algebra because partial fractions is really an algebraic rearranging skill. You may be asked to simplify a rational expression, verify an identity, or prepare a rational function for later techniques. If you recognize that the denominator breaks into distinct linear factors, you know the structure of the answer before you start solving.

This term also helps you avoid the most common setup error: treating every factor the same. A linear factor that appears once gets a constant numerator, while repeated factors need a term for each power. Mixing those up gives the wrong system of equations and wastes time.

When you see a rational expression with a factored denominator, non-repeated linear factors tell you the cleanest path forward. They are the signal that the decomposition will use simple A/(x-a) terms, which makes the algebra much more manageable.

## Connections

### Partial Fraction Decomposition

Non-repeated linear factors are one of the building blocks of partial fraction decomposition. Once you split a rational expression apart, each distinct linear factor gets its own constant-over-linear term. If you can recognize the factor pattern first, setting up the decomposition becomes much faster and less error-prone.

### [Repeated Linear Factors](/college-algebra/key-terms/repeated-linear-factors)

Repeated linear factors look similar, but they need extra terms for each power of the factor. For example, a repeated factor like (x-a)^2 requires both A/(x-a) and B/(x-a)^2. That is why spotting whether a factor repeats changes the entire setup before you even solve for constants.

### [Least Common Denominator (LCD)](/college-algebra/key-terms/common-denominator-lcd)

You usually use the LCD to clear fractions after writing the partial fraction form. For non-repeated linear factors, the LCD is the product of each distinct factor, so multiplying through cancels every denominator cleanly. That leaves an identity you can solve by plugging in values or comparing coefficients.

### [Linear Factors](/college-algebra/key-terms/linear-factors)

A non-repeated linear factor is just a linear factor that appears once. Identifying the factor as linear tells you the numerator should be a constant, not a polynomial. This is the simplest partial-fraction case, and it is the pattern you use before moving to repeated or quadratic factors.

## On the AP Exam

A quiz or problem-set question will usually give you a rational expression and ask you to decompose it, find unknown coefficients, or rewrite the expression in simpler pieces. Your job is to identify which denominator factors are non-repeated, then write one A/(x-a) term for each one. After that, you clear denominators and solve for the constants by substitution or matching coefficients.

If the denominator has only distinct linear factors, you should not add extra powers or more complicated numerators. A fast self-check is that each factor appears once and stays first degree. If your setup includes a term that does not match that pattern, the decomposition is probably off.

## Non-Repeated Linear Factors vs Repeated Linear Factors

Non-repeated linear factors appear once, so each one gets a single constant-over-linear term. Repeated linear factors show up more than once, and you need a separate term for each power of the repeated factor. The setup looks similar, but the decomposition changes as soon as a factor repeats.

## Key Takeaways

- A non-repeated linear factor is a first-degree factor in the denominator that appears only once.
- In partial fractions, each non-repeated linear factor gets one term of the form A/(x-a).
- You usually find the unknown constants by clearing denominators and solving the resulting equation.
- If a factor repeats, the decomposition changes, so you need to check the denominator carefully before writing terms.
- Recognizing non-repeated linear factors makes partial fraction decomposition much more direct.

## FAQs

### What is Non-Repeated Linear Factors in College Algebra?

Non-repeated linear factors are distinct first-degree denominator factors that appear once, such as (x-3) or (x+2). In partial fraction decomposition, each one becomes a simple term like A/(x-3). That setup is the starting point for solving for the unknown coefficients.

### How do you know if a linear factor is non-repeated?

Check the denominator after factoring. If the same linear factor shows up only one time, it is non-repeated. If you see the factor again as a power, like (x-1)^2, then it is repeated and you need more than one term in the decomposition.

### What does a partial fraction term look like for a non-repeated linear factor?

It looks like A/(x-a), where x-a is the linear factor and A is a constant. You do not put a variable in the numerator for this case. The constant is found later by clearing denominators or using substitution.

### What is the difference between non-repeated and repeated linear factors?

Non-repeated linear factors show up once, so they each get one simple fraction term. Repeated linear factors appear more than once, so you need a separate term for each power of that factor. That difference changes the setup of the entire decomposition.

## Related Study Guides

- [11.4 Partial Fractions](/college-algebra/unit-11/4-partial-fractions/study-guide/NaLVhKGfiw59OL79)

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