Algebraic Fractions

What you'll learn

How algebraic fractions work like ordinary fractions.
How to simplify them by factorising and cancelling.
How to divide algebraic fractions safely.
How to solve equations and ratio problems involving algebraic fractions.

1. What is an algebraic fraction?

An algebraic fraction is just a fraction where the numerator, the denominator, or both contain algebra.

The numerator is the top of a fraction. The denominator is the bottom of a fraction.

Definition

Algebraic fraction

An algebraic fraction is a fraction made from algebraic expressions, such as $x+3x−2\frac{x+3}{x-2}$ or $x2+5xx2−4\frac{x^2+5x}{x^2-4}$ .

The big idea is that algebraic fractions follow the same rules as number fractions: you simplify by cancelling common factors, not random terms.

Definition

Factor

A factor is something being multiplied. For example, in $3 (x + 2)$ , the factors are 3 and $x + 2$ .

Key Idea

Only cancel factors

You can cancel a bracket or expression only when it is multiplying the whole numerator and the whole denominator.

Common factors can be cancelled only when the whole numerator and denominator are written as products.

Example

Simplifying by cancelling a common factor

Simplify fully $x2+6xx2+8x+12\frac{x^2+6x}{x^2+8x+12}$ .

The numerator and denominator are factorised so the common factor x + 6 can be cancelled.

Factorise the numerator by taking out the common factor $x$ :
$x^2+6x=x(x+6)$
Factorise the denominator by finding two numbers that multiply to 12 and add to 8:
$x^2+8x+12=(x+2)(x+6)$
Rewrite the fraction using these factors:
$x2+6xx2+8x+12=x(x+6)(x+2)(x+6)\frac{x^2+6x}{x^2+8x+12}=\frac{x(x+6)}{(x+2)(x+6)}$
Cancel the common factor $x + 6$ :
$xx+2\frac{x}{x+2}$

Common Mistake

Cancelling terms

Do not cancel part of a sum. In $x+4x2−16\frac{x+4}{x^2-16}$ , you cannot cancel the $x$ with $x^2$ or the 4 with 16. Factorise first.

2. Factorising first

Factorising means rewriting an expression as a product, which means a multiplication.

You will often need these three patterns:

Common factor: $3x^2+9x=3x(x+3)$
Quadratic factorising: $x^2+7x+10=(x+5)(x+2)$
Difference of two squares: $x^2-25=(x-5)(x+5)$

Definition

Difference of two squares

A difference of two squares has the form $a^2-b^2$ and factorises to $(a - b) (a + b)$ .

Example

Using a difference of two squares

Simplify fully $4x2+16xx2−16\frac{4x^2+16x}{x^2-16}$ .

The denominator x² − 16 is a difference of two squares, giving a common factor x + 4 to cancel.

Factorise the numerator by taking out the common factor $4 x$ :
$4x^2+16x=4x(x+4)$
Factorise the denominator as a difference of two squares:
$x^2-16=(x-4)(x+4)$
Rewrite the fraction:
$4x2+16xx2−16=4x(x+4)(x−4)(x+4)\frac{4x^2+16x}{x^2-16}=\frac{4x(x+4)}{(x-4)(x+4)}$
Cancel the common factor $x + 4$ :
$4xx−4\frac{4x}{x-4}$

Common Mistake

Cancelled factors still mattered originally

If a factor in the denominator is cancelled, the original denominator still could not be zero. In most simplifying questions you give the simplified fraction, but remember this when solving equations.

3. Writing an algebraic fraction in a requested form

Sometimes a question asks you to write the answer in a form like $ax+bx+c\frac{ax+b}{x+c}$ .

Here, $a$ , $b$ , and $c$ are integers, meaning whole numbers, including negative numbers and zero.

The method is still the same: factorise, cancel common factors, then match the form.

Example

Writing in the form requested

Write $2x2+11x+5x2+6x+5\frac{2x^2+11x+5}{x^2+6x+5}$ in the form $ax+bx+c\frac{ax+b}{x+c}$ .

After factorising both quadratics, the shared factor x + 5 cancels and the remaining expression matches the requested form.

Factorise the numerator:
$2x^2+11x+5=(2x+1)(x+5)$
Factorise the denominator:
$x^2+6x+5=(x+1)(x+5)$
Rewrite the fraction:
$2x2+11x+5x2+6x+5=(2x+1)(x+5)(x+1)(x+5)\frac{2x^2+11x+5}{x^2+6x+5}=\frac{(2x+1)(x+5)}{(x+1)(x+5)}$
Cancel the common factor $x + 5$ :
$2x+1x+1\frac{2x+1}{x+1}$
Match it to $ax+bx+c\frac{ax+b}{x+c}$ , so $a = 2$ , $b = 1$ , and $c = 1$ .

Tip

Check your factorising

After factorising, quickly expand your brackets in your head. If they do not return to the original expression, fix the factorising before cancelling.

4. Dividing algebraic fractions

To divide by a fraction, multiply by its reciprocal.

Definition

Reciprocal

The reciprocal of a fraction is the fraction turned upside down. For example, the reciprocal of $x+1x−3\frac{x+1}{x-3}$ is $x−3x+1\frac{x-3}{x+1}$ .

Key Idea

Keep, change, flip

Keep the first fraction, change division to multiplication, and flip the second fraction.

Example

Dividing algebraic fractions

Simplify fully $2x+4x−3÷x2+5x+6x2−3x\frac{2x+4}{x-3}\div\frac{x^2+5x+6}{x^2-3x}$ .

$Dividing algebraic fractions means keeping the first fraction, changing to multiplication, and flipping the second fraction before cancelling factors.$

Factorise everything first:
$2 x + 4 = 2 (x + 2)$ $x^2+5x+6=(x+2)(x+3)$ $x^2-3x=x(x-3)$
Rewrite the division as multiplication by the reciprocal:
$2(x+2)x−3×x(x−3)(x+2)(x+3)\frac{2(x+2)}{x-3}\times\frac{x(x-3)}{(x+2)(x+3)}$
Cancel the common factors $x + 2$ and $x - 3$ :
$2xx+3\frac{2x}{x+3}$

Common Mistake

Forgetting to flip

When dividing algebraic fractions, only the second fraction is flipped. The first fraction stays exactly where it is.

5. Solving equations with algebraic fractions

An equation is a statement that two expressions are equal. When fractions are involved, the usual aim is to remove the denominators.

A common denominator is an expression that all the denominators divide into. Multiplying every term by the common denominator clears the fractions.

Example

Solving an algebraic fraction equation

Solve $3x+2+4x+7=1\frac{3}{x+2}+\frac{4}{x+7}=1$ .

$Multiplying every term by the common denominator (x + 2)(x + 7) clears both fractions.$

Exclude values that make denominators zero:
$x≠−2,x≠−7x\neq -2,\quad x\neq -7$
Multiply every term by the common denominator $(x + 2) (x + 7)$ :
$3 (x + 7) + 4 (x + 2) = (x + 2) (x + 7)$
Expand both sides:
$3x+21+4x+8=x^2+9x+14$
Collect all terms on one side:
$0=x^2+2x-15$
Factorise and solve:
$(x + 5) (x - 3) = 0$ $x=−5orx=3x=-5\quad\text{or}\quad x=3$
Check neither answer was excluded, so the solutions are $x = - 5$ and $x = 3$ .

Tip

Multiply every term

If an equation has three terms, all three must be multiplied by the common denominator — not just the fractions on one side.

6. Ratio statements as fractions

A ratio compares two quantities. A statement like $A : B = C : D$ can be rewritten as $AB=CD\frac{A}{B}=\frac{C}{D}$ .

Then you can cross multiply, which means multiplying each numerator by the opposite denominator.

Example

Solving a ratio equation

Given that $x + 4 : x + 1 = x + 8 : 3 x + 2$ , find the possible values of $x$ .

$The ratio statement is rewritten as equal fractions, then cross multiplication pairs opposite numerator and denominator.$

Rewrite the ratio as a fraction equation:
$x+4x+1=x+83x+2\frac{x+4}{x+1}=\frac{x+8}{3x+2}$
Exclude values that make denominators zero:
$x≠−1,x≠−23x\neq -1,\quad x\neq -\frac{2}{3}$
Cross multiply:
$(x + 4) (3 x + 2) = (x + 8) (x + 1)$
Expand both sides:
$3x^2+14x+8=x^2+9x+8$
Collect terms and factorise:
$2x^2+5x=0$ $x (2 x + 5) = 0$
Solve and check the excluded values:
$x=0orx=−52x=0\quad\text{or}\quad x=-\frac{5}{2}$

Exam technique

In the exam

Factorise every numerator and denominator before you cancel anything.
Cancel only common factors, especially brackets such as $x + 3$ .
When solving, write down excluded values first, then multiply every term by the common denominator.

Self review

Check yourself

Can you explain why $x+4x2−16\frac{x+4}{x^2-16}$ simplifies only after factorising the denominator?
When dividing two algebraic fractions, which fraction do you flip?
In an equation like $5x−1+2x+3=1\frac{5}{x-1}+\frac{2}{x+3}=1$ , what values of $x$ must be excluded before solving?

Algebraic Fractions

What you'll learn

How algebraic fractions work like ordinary fractions.
How to simplify them by factorising and cancelling.
How to divide algebraic fractions safely.
How to solve equations and ratio problems involving algebraic fractions.

1. What is an algebraic fraction?

An algebraic fraction is just a fraction where the numerator, the denominator, or both contain algebra.

The numerator is the top of a fraction. The denominator is the bottom of a fraction.

Definition

Algebraic fraction

An algebraic fraction is a fraction made from algebraic expressions, such as $x+3x−2\frac{x+3}{x-2}$ or $x2+5xx2−4\frac{x^2+5x}{x^2-4}$ .

The big idea is that algebraic fractions follow the same rules as number fractions: you simplify by cancelling common factors, not random terms.

Definition

Factor

A factor is something being multiplied. For example, in $3 (x + 2)$ , the factors are 3 and $x + 2$ .

Key Idea

Only cancel factors

You can cancel a bracket or expression only when it is multiplying the whole numerator and the whole denominator.

Common factors can be cancelled only when the whole numerator and denominator are written as products.

Example

Simplifying by cancelling a common factor

Simplify fully $x2+6xx2+8x+12\frac{x^2+6x}{x^2+8x+12}$ .

The numerator and denominator are factorised so the common factor x + 6 can be cancelled.

Factorise the numerator by taking out the common factor $x$ :
$x^2+6x=x(x+6)$
Factorise the denominator by finding two numbers that multiply to 12 and add to 8:
$x^2+8x+12=(x+2)(x+6)$
Rewrite the fraction using these factors:
$x2+6xx2+8x+12=x(x+6)(x+2)(x+6)\frac{x^2+6x}{x^2+8x+12}=\frac{x(x+6)}{(x+2)(x+6)}$
Cancel the common factor $x + 6$ :
$xx+2\frac{x}{x+2}$

Common Mistake

Cancelling terms

Do not cancel part of a sum. In $x+4x2−16\frac{x+4}{x^2-16}$ , you cannot cancel the $x$ with $x^2$ or the 4 with 16. Factorise first.

2. Factorising first

Factorising means rewriting an expression as a product, which means a multiplication.

You will often need these three patterns:

Common factor: $3x^2+9x=3x(x+3)$
Quadratic factorising: $x^2+7x+10=(x+5)(x+2)$
Difference of two squares: $x^2-25=(x-5)(x+5)$

Definition

Difference of two squares

A difference of two squares has the form $a^2-b^2$ and factorises to $(a - b) (a + b)$ .

Example

Using a difference of two squares

Simplify fully $4x2+16xx2−16\frac{4x^2+16x}{x^2-16}$ .

The denominator x² − 16 is a difference of two squares, giving a common factor x + 4 to cancel.

Factorise the numerator by taking out the common factor $4 x$ :
$4x^2+16x=4x(x+4)$
Factorise the denominator as a difference of two squares:
$x^2-16=(x-4)(x+4)$
Rewrite the fraction:
$4x2+16xx2−16=4x(x+4)(x−4)(x+4)\frac{4x^2+16x}{x^2-16}=\frac{4x(x+4)}{(x-4)(x+4)}$
Cancel the common factor $x + 4$ :
$4xx−4\frac{4x}{x-4}$

Common Mistake

Cancelled factors still mattered originally

3. Writing an algebraic fraction in a requested form

Sometimes a question asks you to write the answer in a form like $ax+bx+c\frac{ax+b}{x+c}$ .

Here, $a$ , $b$ , and $c$ are integers, meaning whole numbers, including negative numbers and zero.

The method is still the same: factorise, cancel common factors, then match the form.

Example

Writing in the form requested

Write $2x2+11x+5x2+6x+5\frac{2x^2+11x+5}{x^2+6x+5}$ in the form $ax+bx+c\frac{ax+b}{x+c}$ .

After factorising both quadratics, the shared factor x + 5 cancels and the remaining expression matches the requested form.

Factorise the numerator:
$2x^2+11x+5=(2x+1)(x+5)$
Factorise the denominator:
$x^2+6x+5=(x+1)(x+5)$
Rewrite the fraction:
$2x2+11x+5x2+6x+5=(2x+1)(x+5)(x+1)(x+5)\frac{2x^2+11x+5}{x^2+6x+5}=\frac{(2x+1)(x+5)}{(x+1)(x+5)}$
Cancel the common factor $x + 5$ :
$2x+1x+1\frac{2x+1}{x+1}$
Match it to $ax+bx+c\frac{ax+b}{x+c}$ , so $a = 2$ , $b = 1$ , and $c = 1$ .

Tip

Check your factorising

After factorising, quickly expand your brackets in your head. If they do not return to the original expression, fix the factorising before cancelling.

4. Dividing algebraic fractions

To divide by a fraction, multiply by its reciprocal.

Definition

Reciprocal

The reciprocal of a fraction is the fraction turned upside down. For example, the reciprocal of $x+1x−3\frac{x+1}{x-3}$ is $x−3x+1\frac{x-3}{x+1}$ .

Key Idea

Keep, change, flip

Keep the first fraction, change division to multiplication, and flip the second fraction.

Example

Dividing algebraic fractions

Simplify fully $2x+4x−3÷x2+5x+6x2−3x\frac{2x+4}{x-3}\div\frac{x^2+5x+6}{x^2-3x}$ .

$Dividing algebraic fractions means keeping the first fraction, changing to multiplication, and flipping the second fraction before cancelling factors.$

Factorise everything first:
$2 x + 4 = 2 (x + 2)$ $x^2+5x+6=(x+2)(x+3)$ $x^2-3x=x(x-3)$
Rewrite the division as multiplication by the reciprocal:
$2(x+2)x−3×x(x−3)(x+2)(x+3)\frac{2(x+2)}{x-3}\times\frac{x(x-3)}{(x+2)(x+3)}$
Cancel the common factors $x + 2$ and $x - 3$ :
$2xx+3\frac{2x}{x+3}$

Common Mistake

Forgetting to flip

When dividing algebraic fractions, only the second fraction is flipped. The first fraction stays exactly where it is.

5. Solving equations with algebraic fractions

An equation is a statement that two expressions are equal. When fractions are involved, the usual aim is to remove the denominators.

A common denominator is an expression that all the denominators divide into. Multiplying every term by the common denominator clears the fractions.

Example

Solving an algebraic fraction equation

Solve $3x+2+4x+7=1\frac{3}{x+2}+\frac{4}{x+7}=1$ .

$Multiplying every term by the common denominator (x + 2)(x + 7) clears both fractions.$

Exclude values that make denominators zero:
$x≠−2,x≠−7x\neq -2,\quad x\neq -7$
Multiply every term by the common denominator $(x + 2) (x + 7)$ :
$3 (x + 7) + 4 (x + 2) = (x + 2) (x + 7)$
Expand both sides:
$3x+21+4x+8=x^2+9x+14$
Collect all terms on one side:
$0=x^2+2x-15$
Factorise and solve:
$(x + 5) (x - 3) = 0$ $x=−5orx=3x=-5\quad\text{or}\quad x=3$
Check neither answer was excluded, so the solutions are $x = - 5$ and $x = 3$ .

Tip

Multiply every term

If an equation has three terms, all three must be multiplied by the common denominator — not just the fractions on one side.

6. Ratio statements as fractions

A ratio compares two quantities. A statement like $A : B = C : D$ can be rewritten as $AB=CD\frac{A}{B}=\frac{C}{D}$ .

Then you can cross multiply, which means multiplying each numerator by the opposite denominator.

Example

Solving a ratio equation

Given that $x + 4 : x + 1 = x + 8 : 3 x + 2$ , find the possible values of $x$ .

$The ratio statement is rewritten as equal fractions, then cross multiplication pairs opposite numerator and denominator.$

Rewrite the ratio as a fraction equation:
$x+4x+1=x+83x+2\frac{x+4}{x+1}=\frac{x+8}{3x+2}$
Exclude values that make denominators zero:
$x≠−1,x≠−23x\neq -1,\quad x\neq -\frac{2}{3}$
Cross multiply:
$(x + 4) (3 x + 2) = (x + 8) (x + 1)$
Expand both sides:
$3x^2+14x+8=x^2+9x+8$
Collect terms and factorise:
$2x^2+5x=0$ $x (2 x + 5) = 0$
Solve and check the excluded values:
$x=0orx=−52x=0\quad\text{or}\quad x=-\frac{5}{2}$

Exam technique

In the exam

Factorise every numerator and denominator before you cancel anything.
Cancel only common factors, especially brackets such as $x + 3$ .
When solving, write down excluded values first, then multiply every term by the common denominator.

Self review

Check yourself

Can you explain why $x+4x2−16\frac{x+4}{x^2-16}$ simplifies only after factorising the denominator?
When dividing two algebraic fractions, which fraction do you flip?
In an equation like $5x−1+2x+3=1\frac{5}{x-1}+\frac{2}{x+3}=1$ , what values of $x$ must be excluded before solving?

Algebraic Fractions

What you'll learn

1. What is an algebraic fraction?

2. Factorising first

3. Writing an algebraic fraction in a requested form

4. Dividing algebraic fractions

5. Solving equations with algebraic fractions

6. Ratio statements as fractions

Recap questions

You've reached the end

Algebraic Fractions

What you'll learn

1. What is an algebraic fraction?

2. Factorising first

3. Writing an algebraic fraction in a requested form

4. Dividing algebraic fractions

5. Solving equations with algebraic fractions

6. Ratio statements as fractions

Recap questions

You've reached the end

Algebraic Fractions

What you'll learn

1. What is an algebraic fraction?

2. Factorising first

3. Writing an algebraic fraction in a requested form

4. Dividing algebraic fractions

5. Solving equations with algebraic fractions

6. Ratio statements as fractions

Recap questions

You've reached the end

Algebraic Fractions Revision Guide

Algebraic Fractions

What you'll learn

1. What is an algebraic fraction?

2. Factorising first

3. Writing an algebraic fraction in a requested form

4. Dividing algebraic fractions

5. Solving equations with algebraic fractions

6. Ratio statements as fractions

Recap questions

You've reached the end

Algebraic Fractions Revision Guide