Learning Objectives
  • Simplify algebraic fractions by factorising and cancelling common factors.
  • Multiply, divide, add and subtract algebraic fractions correctly.
  • State restrictions that prevent a denominator from being zero.
  • Substitute values into formulas accurately, including negative and fractional values.
  • Change the subject of a formula, including cases where the subject appears twice or is under a power or root.
Key Terms
Algebraic fraction
A fraction whose numerator, denominator or both contain algebraic expressions.
Rational expression
An algebraic expression written as one polynomial divided by another.
Common factor
A factor shared by all relevant terms or expressions.
Restriction
A value excluded because it would make a denominator zero or an operation undefined.
Formula
An equation that shows a relationship between quantities.
Subject of a formula
The variable written alone on one side of the formula.
Lowest common denominator
The smallest expression that is a multiple of every denominator in a set of fractions.
What This Chapter Covers
  • Factorising before cancellation
  • Four operations with algebraic fractions
  • Domain restrictions created by denominators
  • Substitution into formulas
  • Changing the subject in simple and multi-step formulas
Algebraic Fractions Are Fractions Of Factors

The same fraction laws used with numbers also apply to algebraic fractions. The key difference is that algebraic expressions often need to be factorised before a common factor becomes visible. Cancellation is valid only between factors that multiply the whole numerator and denominator. It is not valid to cancel separate terms joined by addition or subtraction.

For example, in \frac{6x^2y}{9xy^2}, the numerator and denominator are products. Common factors can be cancelled. In \frac{x+3}{x}, the x in the denominator is not a factor of the whole numerator, so it cannot be cancelled.

\frac{6x^2y}{9xy^2}=\frac{2x}{3y}

Cancel the common numerical factor 3, one factor of x, and one factor of y.

Restrictions On Values

A denominator may never equal zero. Therefore, every algebraic fraction carries restrictions. In \frac{5}{x-4}, x\ne4. In \frac{x+1}{(x-2)(x+5)}, x\ne2 and x\ne-5. A factor may disappear after simplification, but its original restriction still remains because the original expression was undefined at that value.

Worked Example: Simplify And State Restrictions

Question: Simplify \frac{x^2-9}{x^2+x-6} and state the excluded values.

  1. Factorise the numerator: x^2-9=(x-3)(x+3).
  2. Factorise the denominator: x^2+x-6=(x+3)(x-2).
  3. Cancel the common factor (x+3).
  4. The original denominator is zero when x=-3 or x=2.

Answer: \frac{x-3}{x-2}, where x\ne-3,2.

Multiplying Algebraic Fractions

Factorise every numerator and denominator, then cancel common factors before multiplying. Early cancellation keeps the arithmetic small and reduces errors. The restrictions come from all original denominators.

Worked Example: Multiplication

Question: Simplify \frac{x^2-4}{3x}\times\frac{9x^2}{x+2}.

  1. Factorise x^2-4=(x-2)(x+2).
  2. Write \frac{(x-2)(x+2)}{3x}\times\frac{9x^2}{x+2}.
  3. Cancel (x+2), cancel one x, and simplify 9/3 to 3.

Answer: 3x(x-2), with x\ne0,-2.

Dividing Algebraic Fractions

To divide by a fraction, multiply by its reciprocal. The divisor must not be zero, so a numerator in the divisor can create an additional restriction.

\frac{A}{B}\div\frac{C}{D}=\frac{A}{B}\times\frac{D}{C}
Worked Example: Division

Question: Simplify \frac{4x}{x^2-1}\div\frac{2x^2}{x+1}.

  1. Change division to multiplication by the reciprocal.
  2. Factorise x^2-1=(x-1)(x+1).
  3. Write \frac{4x}{(x-1)(x+1)}\times\frac{x+1}{2x^2}.
  4. Cancel x+1, one factor of x, and simplify 4/2.

Answer: \frac{2}{x(x-1)}, with x\ne-1,0,1.

Adding And Subtracting Algebraic Fractions

Fractions can be added or subtracted only after they have a common denominator. Use the lowest common denominator whenever possible. Multiply both the numerator and denominator of each fraction by the factor needed to create that denominator. Then combine the numerators, expand if necessary, collect like terms, and factorise again to check whether the final result simplifies.

Worked Example: Unlike Denominators

Question: Simplify \frac{2}{x-1}+\frac{3}{x+2}.

  1. The lowest common denominator is (x-1)(x+2).
  2. Rewrite as \frac{2(x+2)+3(x-1)}{(x-1)(x+2)}.
  3. Expand the numerator: 2x+4+3x-3=5x+1.

Answer: \frac{5x+1}{(x-1)(x+2)}, where x\ne1,-2.

Worked Example: Subtraction With A Repeated Factor

Question: Simplify \frac{1}{x}-\frac{2}{x^2}.

  1. The lowest common denominator is x^2.
  2. Rewrite \frac{1}{x}=\frac{x}{x^2}.
  3. Subtract the numerators: \frac{x-2}{x^2}.

Answer: \frac{x-2}{x^2}, where x\ne0.

Substitution Into Formulas

When substituting a negative number, place it in brackets. When a formula contains powers, calculate the power before multiplying. Keep exact values such as fractions and surds until the final line unless a decimal approximation is requested.

Worked Example: Accurate Substitution

Question: Given P=2a^2-3ab+b^2, find P when a=-2 and b=3.

  1. Substitute using brackets: P=2(-2)^2-3(-2)(3)+3^2.
  2. Calculate powers: 2(4)+18+9.
  3. Add the terms.

Answer: P=35.

Changing The Subject Of A Formula

Changing the subject means rearranging the formula so that the required variable is alone. Use inverse operations in a logical order and preserve equality by performing the same operation on both sides.

A Reliable Rearrangement Method
  1. Identify every operation currently applied to the required subject.
  2. Undo addition and subtraction before multiplication and division when appropriate.
  3. Collect all terms containing the subject on one side if it appears more than once.
  4. Factorise the subject out of those terms.
  5. Divide by the remaining factor.
  6. If the subject is squared, cubed or under a root, apply the inverse power or root at the correct stage.
Worked Example: Subject Appears Once

Question: Make r the subject of A=\pi r^2.

  1. Divide both sides by \pi: r^2=A/\pi.
  2. Take the positive square root because a radius is positive.

Answer: r=\sqrt{\frac{A}{\pi}}.

Worked Example: Subject Appears Twice

Question: Make x the subject of y=ax+bx-c.

  1. Add c: y+c=ax+bx.
  2. Factorise x: y+c=x(a+b).
  3. Divide by a+b.

Answer: x=\frac{y+c}{a+b}, provided a+b\ne0.

Worked Example: Fractional Formula

Question: Make h the subject of V=\frac13\pi r^2h.

  1. Multiply both sides by 3: 3V=\pi r^2h.
  2. Divide by \pi r^2.

Answer: h=\frac{3V}{\pi r^2}.

Checking A Rearranged Formula

Substitute simple numbers into both the original and rearranged formulas. If both produce the same result, the rearrangement is likely correct. Also check dimensions informally: if the original subject represents a length, the rearranged expression should behave like a length rather than an area or volume.

Examination Guidance
  • Factorise fully before cancelling; Cambridge expects the final rational expression in simplest form.
  • State restrictions when the question asks for values for which an expression is undefined.
  • Use brackets around substituted negative values.
  • When changing the subject, show one valid algebraic operation per line.
  • After adding fractions, factorise the final numerator and denominator again before deciding that no cancellation is possible.
Common Mistakes
  • Cancelling terms across addition, such as cancelling x in (x+2)/x.
  • Forgetting that a cancelled factor still contributes an excluded value from the original expression.
  • Inverting the first fraction instead of the divisor when dividing fractions.
  • Changing signs incorrectly when removing brackets after subtraction.
  • Taking only a square root without considering the context; algebraically x^2=k gives x=\pm\sqrt{k}, but a physical length may require the positive value.
Knowledge Check And Practice

1. Simplify \frac{12a^3b}{18a^2b^2}.

Answer: \frac{2a}{3b}, where a\ne0 and b\ne0 in the original expression.

2. Simplify \frac{x^2-16}{x^2-8x+16}.

Answer: \frac{x+4}{x-4}, with x\ne4.

3. Simplify \frac{3}{x}+\frac{2}{x-1}.

Answer: \frac{5x-3}{x(x-1)}.

4. Simplify \frac{x}{x+2}\div\frac{x^2}{x^2-4}.

Answer: \frac{x-2}{x}, with x\ne-2,0,2.

5. Make t the subject of v=u+at.

Answer: t=\frac{v-u}{a}.

6. Make x the subject of p=\frac{x-q}{r}.

Answer: x=pr+q.

7. Make m the subject of E=mc^2.

Answer: m=\frac{E}{c^2}.

8. Make x the subject of y=\sqrt{3x+1}.

Answer: x=\frac{y^2-1}{3}.
PreviousNext