Factorization and Algebraic Fractions

Mathematics · IB Middle Years Programme (MYP) · 10 min read · Updated: 23 Oct 2026

Welcome to Factorization and Algebraic Fractions!

Have you ever taken something apart to see how it works, like taking apart a mechanical pen or unzipping a backpack to unpack its contents neatly? In mathematics, factorization is doing exactly that. While expanding brackets lets you unpack and multiply things out, factorizing allows you to do the reverse: break an expression down into its building blocks (its factors).

Once you master factoring, working with algebraic fractions (fractions that have variables like \(x\) in the numerator or denominator) becomes as natural as working with regular numbers. These skills are essential in the Thinking with models branch of the MYP, allowing you to model real-world rates, streamline complex formulas, and communicate your reasoning clearly for Criterion A and Criterion C.

Note: For solving full quadratic equations, take a look at our companion chapter "Quadratic Expressions and Equations". Here, we will focus on factoring expressions and mastering algebraic fractions at both Standard and Extended levels.

Part 1: The Foundations of Factorization

To factorize an algebraic expression means to write it as a product of two or more simpler terms or expressions. It is the direct opposite of expanding brackets.

Think of it this way:
Expanding: \(3(x + 4) = 3x + 12\)
Factorizing: \(3x + 12 = 3(x + 4)\)

Method 1: Taking Out the Highest Common Factor (HCF)

Always look for this first! Look at each term in the expression and find the largest number and variable parts that divide evenly into all terms.

Worked Example 1: Factorize \(6x^2 + 9x\)
Step 1: Find the HCF of the numbers \(6\) and \(9\). Both are divisible by \(3\).
Step 2: Find the HCF of the variable parts \(x^2\) and \(x\). Both contain at least one factor of \(x\).
Step 3: The overall HCF is \(3x\).
Step 4: Divide each original term by \(3x\) to see what remains inside the bracket:
\(6x^2 \div 3x = 2x\)
\(9x \div 3x = 3\)
Final Result: \(6x^2 + 9x = 3x(2x + 3)\)

Quick Check Tip: You can always check your answer by expanding your brackets back out. If you get your starting expression, your factorization is correct!

Method 2: Factoring by Grouping (Four Terms)

When you see an expression with four terms, there might not be a single factor common to all four. Instead, group them in pairs.

Worked Example 2: Factorize \(ax + ay + 2x + 2y\)
Step 1: Group the first two terms and the last two terms: \((ax + ay) + (2x + 2y)\).
Step 2: Take the HCF out of each pair separately:
\(a(x + y) + 2(x + y)\)
Step 3: Notice that the bracket \((x + y)\) is now a common factor shared by both parts!
Step 4: Factor out the bracket \((x + y)\):
Final Result: \((x + y)(a + 2)\)

Common Trap to Avoid: Be careful with negative signs when grouping. For example, in \(x^2 - 3x - 2x + 6\), grouping the last two terms means factoring out \(-2\), giving \(-2(x - 3)\), not \(-2(x + 3)\).

Part 2: Special Factorization Patterns

Recognizing patterns helps you factorize quickly and accurately. Two of the most useful patterns in algebra are the Difference of Two Squares and Quadratic Trinomials.

The Difference of Two Squares (DOTS)

Whenever you see a square term minus another square term, use the identity:
\(a^2 - b^2 = (a - b)(a + b)\)

Memory Trick: Look for two terms, a minus sign between them, and perfect squares on both ends.

Worked Example 3: Factorize \(x^2 - 25\)
Here, \(a = x\) and \(b = 5\) (since \(5^2 = 25\)).
Therefore, \(x^2 - 25 = (x - 5)(x + 5)\).

Worked Example 4: Factorize \(4x^2 - 49y^2\)
Notice that \(4x^2 = (2x)^2\) and \(49y^2 = (7y)^2\).
Therefore, \(4x^2 - 49y^2 = (2x - 7y)(2x + 7y)\).

Factoring Quadratic Trinomials of the Form \(x^2 + bx + c\)

A trinomial has three terms. When the coefficient of \(x^2\) is \(1\), we want to find two numbers that multiply to give \(c\) and add to give \(b\).

Worked Example 5: Factorize \(x^2 + 7x + 12\)
Step 1: List pairs of numbers that multiply to \(12\):
\(1 \times 12 = 12\) (sum: \(1 + 12 = 13\))
\(2 \times 6 = 12\) (sum: \(2 + 6 = 8\))
\(3 \times 4 = 12\) (sum: \(3 + 4 = 7\)) — This is our pair!
Step 2: Write the factors directly:
Final Result: \(x^2 + 7x + 12 = (x + 3)(x + 4)\)

Worked Example 6 (Negative Signs): Factorize \(x^2 - 5x - 14\)
We need two numbers that multiply to \(-14\) and add to \(-5\).
Consider \(-7\) and \(+2\):
\((-7) \times 2 = -14\)
\((-7) + 2 = -5\)
Final Result: \(x^2 - 5x - 14 = (x - 7)(x + 2)\)

Extended Level: Trinomials with \(ax^2 + bx + c\) where \(a \neq 1\)

When the coefficient in front of \(x^2\) is greater than \(1\), use the ac method (splitting the middle term).

Worked Example 7: Factorize \(2x^2 + 7x + 3\)
Step 1: Multiply \(a\) and \(c\): \(a \times c = 2 \times 3 = 6\).
Step 2: Find two numbers that multiply to \(6\) and add to \(b = 7\). The numbers are \(6\) and \(1\).
Step 3: Split the middle term \(7x\) into \(6x + 1x\):
\(2x^2 + 6x + x + 3\)
Step 4: Factorize by grouping in pairs:
\(2x(x + 3) + 1(x + 3)\)
Final Result: \((x + 3)(2x + 1)\)

Key Takeaway for Factorization: Always check for a common numerical or variable factor first! Then ask yourself: Is it a difference of two squares? A simple trinomial? Or four terms to group?

Part 3: Introduction to Algebraic Fractions

An algebraic fraction (or rational expression) is a fraction where the numerator, the denominator, or both contain algebraic terms. For example: \(\frac{3}{x}\), \(\frac{x + 2}{5}\), or \(\frac{x - 1}{x^2 - 4}\).

The Fundamental Rule: Division by Zero is Undefined

In mathematics, the denominator of a fraction can never be zero. Values of a variable that make the denominator equal to zero are called non-permissible values or excluded values.

For example, in the fraction \(\frac{5}{x - 3}\), the denominator cannot be zero:
\(x - 3 \neq 0 \implies x \neq 3\)
Stating these restrictions demonstrates high-level communication (Criterion C).

Simplifying Algebraic Fractions

To simplify an algebraic fraction, you must factorize first, and then cancel common factors shared by the entire numerator and denominator.

Crucial Rule: You can ONLY cancel factors (things being multiplied), NEVER individual terms (things being added or subtracted)!
Wrong: In \(\frac{x + 4}{4}\), you cannot cancel the \(4\)s. The answer is not \(x + 1\).
Right: In \(\frac{4(x + 1)}{4}\), the factor \(4\) cancels cleanly, leaving \(x + 1\).

Standard Mathematics Example (First Degree):
Simplify \(\frac{4x + 12}{2x + 6}\)
Step 1: Factorize the numerator: \(4(x + 3)\).
Step 2: Factorize the denominator: \(2(x + 3)\).
Step 3: Cancel the common binomial factor \((x + 3)\):
\(\frac{4(x + 3)}{2(x + 3)} = \frac{4}{2} = 2\) (where \(x \neq -3\)).

Extended Mathematics Example (Second Degree Denominators):
Simplify \(\frac{x^2 - 9}{x^2 + 5x + 6}\)
Step 1: Factorize the numerator using DOTS: \(x^2 - 9 = (x - 3)(x + 3)\).
Step 2: Factorize the denominator trinomial: \(x^2 + 5x + 6 = (x + 2)(x + 3)\).
Step 3: Cancel the shared factor \((x + 3)\):
\(\frac{(x - 3)(x + 3)}{(x + 2)(x + 3)} = \frac{x - 3}{x + 2}\) (where \(x \neq -2, -3\)).

Part 4: Operations with Algebraic Fractions

Algebraic fractions follow the exact same rules as numerical fractions: find common denominators for adding and subtracting, and multiply straight across for multiplying.

1. Multiplication and Division

Multiplying: Factorize all numerators and denominators, cancel any common factors, then multiply across.
Dividing: Keep the first fraction, flip the second fraction (find its reciprocal), and multiply!

Worked Example 8 (Multiplying):
Calculate \(\frac{x + 2}{3x} \times \frac{6x^2}{x^2 - 4}\)
Step 1: Factorize everything completely:
\(\frac{x + 2}{3x} \times \frac{6x^2}{(x - 2)(x + 2)}\)
Step 2: Cancel common factors: \((x + 2)\) cancels with \((x + 2)\), and \(\frac{6x^2}{3x}\) simplifies to \(2x\).
Step 3: Combine the remaining pieces:
Final Result: \(\frac{2x}{x - 2}\)

Worked Example 9 (Dividing):
Calculate \(\frac{2x}{x - 1} \div \frac{4x^2}{3(x - 1)}\)
Step 1: Change division to multiplication by inverting the second fraction:
\(\frac{2x}{x - 1} \times \frac{3(x - 1)}{4x^2}\)
Step 2: Cancel common factors: \((x - 1)\) cancels with \((x - 1)\), and \(\frac{2x}{4x^2} = \frac{1}{2x}\).
Final Result: \(\frac{3}{2x}\)

2. Addition and Subtraction

To add or subtract algebraic fractions, they must share a Lowest Common Denominator (LCD).

Standard Mathematics Level (Linear / First Degree Denominators):
Calculate \(\frac{3}{x} + \frac{2}{x + 1}\)
Step 1: The LCD of \(x\) and \(x + 1\) is \(x(x + 1)\).
Step 2: Rewrite each fraction with the common denominator:
\(\frac{3(x + 1)}{x(x + 1)} + \frac{2(x)}{x(x + 1)}\)
Step 3: Expand the numerators and combine over the single denominator:
\(\frac{3x + 3 + 2x}{x(x + 1)} = \frac{5x + 3}{x(x + 1)}\)

Extended Mathematics Level (Quadratic / Second Degree Denominators):
Calculate \(\frac{2}{x - 3} - \frac{5}{x^2 - 9}\)
Step 1: Factorize all denominators to spot the LCD:
\(x^2 - 9 = (x - 3)(x + 3)\)
The LCD is \((x - 3)(x + 3)\).
Step 2: Adjust the first fraction so it has the LCD:
\(\frac{2(x + 3)}{(x - 3)(x + 3)} - \frac{5}{(x - 3)(x + 3)}\)
Step 3: Combine the numerators carefully:
\(\frac{2(x + 3) - 5}{(x - 3)(x + 3)} = \frac{2x + 6 - 5}{(x - 3)(x + 3)} = \frac{2x + 1}{(x - 3)(x + 3)}\)

Common Trap to Avoid: When subtracting, distribute the negative sign across the entire numerator of the second fraction! A missing negative bracket is one of the most common mistakes in on-screen eAssessments.

Part 5: Thinking with Models and Real-Life Contexts

Algebraic fractions are not just abstract symbols; they are powerful models for rates in science, economics, and real-world collaboration (Criterion D).

Real-World Analogy: Shared Work Rate

If a machine can pack an order of medical kits in \(t\) hours, its rate of work is the fraction of the job completed in one hour: \(\frac{1}{t}\).
If a newer machine completes the same job in \(t - 2\) hours, its hourly rate is \(\frac{1}{t - 2}\).
Working together, their combined rate per hour is modelled by adding these algebraic fractions:
\(\text{Combined Rate} = \frac{1}{t} + \frac{1}{t - 2}\)
Using our LCD technique:
\(\frac{1(t - 2)}{t(t - 2)} + \frac{1(t)}{t(t - 2)} = \frac{t - 2 + t}{t(t - 2)} = \frac{2t - 2}{t(t - 2)}\)

By simplifying this model into a single algebraic fraction, engineers can quickly compute the total time needed for any initial machine speed \(t\).

Quick Review and Study Checklist

Before your eAssessment, make sure you can confidently check off each of these core skills:

1. Common Factors: Can you extract numerical and variable HCFs from multi-term expressions?
2. Grouping: Can you group four terms in pairs to find a shared bracket?
3. Difference of Squares: Do you recognize \(a^2 - b^2 = (a - b)(a + b)\) instantly?
4. Trinomials: Can you find factors of \(c\) that add to \(b\) in expressions like \(x^2 + bx + c\)?
5. Simplifying Fractions: Do you always factorize completely before cancelling?
6. Non-Permissible Values: Can you state which values make the denominator zero (e.g. \(x \neq 0\))?
7. Operations: Can you find the LCD to combine fractions with both first degree (Standard) and second degree (Extended) denominators?

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What does it mean to factorize an algebraic expression?

Factorizing means rewriting an algebraic expression as a product of simpler terms or expressions, which is the exact opposite of expanding brackets.

How do you factorize an expression by taking out the Highest Common Factor (HCF)?

Identify the largest numerical and variable factors common to every term, place this HCF outside a bracket, and divide each term by it to determine what remains inside.

When and how should you factorize an algebraic expression using grouping?

Use grouping when an expression has four terms with no single overall common factor by grouping them in pairs, factoring the HCF from each pair, and extracting the resulting shared binomial bracket.

What is the Difference of Two Squares (DOTS) identity and how is it used?

The identity is \(a^2 - b^2 = (a - b)(a + b)\), used whenever an expression consists of one perfect square subtracted from another.

How do you factorize a monic quadratic trinomial of the form \(x^2 + bx + c\)?

Find two numbers that multiply to give the constant term \(c\) and add to give the linear coefficient \(b\), then write the expression as \((x + p)(x + q)\).

How do you factorize a non-monic quadratic trinomial of the form \(ax^2 + bx + c\) where \(a \neq 1\)?

Use the "ac method" by finding two numbers that multiply to \(a \times c\) and add to \(b\), splitting the middle term \(bx\), and factoring by grouping.

What are non-permissible (or excluded) values in algebraic fractions?

They are values of the variable that make any denominator equal to zero, which is mathematically undefined.

What is the golden rule for simplifying algebraic fractions?

Factorize the numerator and denominator completely first, and cancel only common factors, never individual terms.

How do you multiply and divide algebraic fractions?

To multiply, factorize fully, cancel shared factors across numerators and denominators, and multiply across; to divide, invert the divisor fraction (its reciprocal) and multiply.

How do you add or subtract algebraic fractions with different denominators?

Find the lowest common denominator (LCD), rewrite each fraction with this common denominator, and combine their numerators over the single denominator.

How can algebraic fractions be used to model shared work rates in real-life contexts?

If a task takes \(t\) time units, the rate of work per unit time is \(\frac{1}{t}\); combining multiple workers or machines corresponds to adding their individual fractional rates.

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