Section B — Chapters 23–39

Algebraic Fractions

Junior Cycle — 3rd Year

✓By the end of this lesson students will be able to simplify algebraic fractions by factorising and cancelling common factors.
✓By the end of this lesson students will be able to add and subtract algebraic fractions by finding a common denominator.
✓By the end of this lesson students will be able to multiply and divide algebraic fractions.
✓By the end of this lesson students will be able to solve equations involving algebraic fractions.

Key concepts

Algebraic Fraction

An algebraic fraction is a fraction where the numerator and/or the denominator contain algebraic expressions (expressions with variables). Just like numerical fractions, they represent a part of a whole. For example, (x+1)/2 or (x^2 - 4)/(x+2).

Simplifying Algebraic Fractions

To simplify an algebraic fraction, we must first factorise both the numerator and the denominator completely. Once factorised, we can cancel out any common factors that appear in both the numerator and the denominator. It is crucial to remember that you can only cancel common *factors*, not individual terms.

Adding and Subtracting Algebraic Fractions

To add or subtract algebraic fractions, they must have a common denominator. We find the Lowest Common Denominator (LCD) of all the fractions involved. Each fraction is then rewritten with this LCD, and finally, the numerators are added or subtracted, keeping the common denominator.

a/b + c/d = (ad + bc)/bd (This is a general form; finding the LCD is usually more efficient)

Multiplying Algebraic Fractions

To multiply algebraic fractions, we multiply the numerators together and multiply the denominators together. The resulting fraction should then be simplified if possible. Often, it's easier to factorise and cancel common factors *before* performing the multiplication.

(a/b) * (c/d) = (ac)/(bd)

Dividing Algebraic Fractions

To divide by an algebraic fraction, we use the 'invert and multiply' rule. This means we flip the second fraction (the divisor) upside down to find its reciprocal, and then multiply it by the first fraction.

(a/b) / (c/d) = (a/b) * (d/c) = (ad)/(bc)

Solving Equations with Algebraic Fractions

To solve an equation that contains algebraic fractions, the primary goal is to eliminate the denominators. This is achieved by multiplying every single term in the entire equation by the Lowest Common Denominator (LCD) of all the fractions present. This process clears the denominators, leaving a simpler equation (often linear or quadratic) to solve. Always check that your solution does not make any original denominator equal to zero, as this would make the fraction undefined.

Key facts to remember

1To simplify an algebraic fraction, factorise the numerator and denominator fully, then cancel any common factors.
2To add or subtract algebraic fractions, you must first find the Lowest Common Denominator (LCD).
3To multiply algebraic fractions, multiply the numerators together and the denominators together.
4To divide by an algebraic fraction, 'invert and multiply' (multiply by the reciprocal of the second fraction).
5When solving equations with algebraic fractions, multiply every term by the LCD to eliminate the denominators.
6Always be mindful of values that would make a denominator zero, as these values are not permissible for the variable.
7Strong factorising skills (common factor, difference of two squares, quadratic trinomials) are fundamental for working with algebraic fractions.

Worked examples

Example 1

Simplify the algebraic fraction: (x^2 + 5x) / (x^2 - 25)

IStep 1: Factorise the numerator. The common factor is x: x(x + 5)

IIStep 2: Factorise the denominator. This is a difference of two squares: (x - 5)(x + 5)

IIIStep 3: Rewrite the fraction with the factorised numerator and denominator: (x(x + 5)) / ((x - 5)(x + 5))

IVStep 4: Cancel the common factor (x + 5) from the numerator and denominator.

VStep 5: Write the simplified fraction.

Answer

x / (x - 5)

Remember that x cannot be equal to 5 or -5, as these values would make the original denominator zero, rendering the expression undefined.

Example 2

Express as a single fraction: 4/(x+2) - 3/(x-1)

IStep 1: Find the Lowest Common Denominator (LCD) of (x+2) and (x-1). The LCD is (x+2)(x-1).

IIStep 2: Rewrite each fraction with the LCD:

III For 4/(x+2), multiply numerator and denominator by (x-1): (4(x-1)) / ((x+2)(x-1))

IV For 3/(x-1), multiply numerator and denominator by (x+2): (3(x+2)) / ((x+2)(x-1))

VStep 3: Subtract the new numerators, keeping the common denominator:

VI (4(x-1) - 3(x+2)) / ((x+2)(x-1))

VIIStep 4: Expand the brackets in the numerator:

VIII (4x - 4 - 3x - 6) / ((x+2)(x-1))

9Step 5: Combine like terms in the numerator:

10 (x - 10) / ((x+2)(x-1))

Answer

(x - 10) / ((x+2)(x-1))

It is generally best to leave the denominator in factorised form unless specifically asked to expand it.

Example 3

Solve the equation: (x-1)/3 + (x+2)/4 = 5

IStep 1: Find the Lowest Common Denominator (LCD) of the denominators 3 and 4. The LCD is 12.

IIStep 2: Multiply every term in the equation by the LCD (12):

III 12 * ((x-1)/3) + 12 * ((x+2)/4) = 12 * 5

IVStep 3: Simplify each term by cancelling the denominators:

V 4(x-1) + 3(x+2) = 60

VIStep 4: Expand the brackets:

VII 4x - 4 + 3x + 6 = 60

VIIIStep 5: Combine like terms on the left side of the equation:

9 7x + 2 = 60

10Step 6: Subtract 2 from both sides:

11 7x = 58

12Step 7: Divide by 7 to solve for x:

13 x = 58/7

Answer

x = 58/7

Remember to multiply *all* terms by the LCD, including any constant terms that are not fractions.

Common mistakes

✗Cancelling individual terms instead of common factors (e.g., incorrectly simplifying (x+1)/x to 1+1/x or just 1).
✗Making sign errors when expanding brackets, especially when a minus sign precedes a fraction or bracket (e.g., -(x-2) becoming -x-2 instead of -x+2).
✗Not finding the correct Lowest Common Denominator (LCD) when adding or subtracting fractions.
✗Forgetting to multiply *all* terms in an equation by the LCD, particularly constant terms that are not fractions.
✗Not fully factorising expressions before attempting to simplify, leading to missed common factors.

Exam tips

★Always show your full step-by-step working, especially when factorising and finding common denominators.
★When solving equations involving algebraic fractions, your first step should always be to multiply every term by the LCD to clear the denominators.
★Double-check your factorisation carefully before cancelling any terms.
★When adding or subtracting, use brackets around your numerators after finding the common denominator to prevent sign errors.
★After solving an equation, substitute your answer back into the original equation to verify your solution.

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