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Simplify and combine algebraic fractions

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Simplify and combine algebraic fractions

Record the original denominator restrictions and exclude zeros of any divisor. Factorise before simplifying, use a reciprocal for division and a common denominator for addition or subtraction

\[\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c},\quad b,c,d\ne0\]
  1. 1Record the original restrictions Every original denominator must be nonzero. When dividing, the divisor must also be nonzero. Cancelling a factor does not restore an excluded input
  2. 2Factorise first Write each numerator and denominator as products before looking for common factors \(\displaystyle \frac{6x\left(x+2\right)}{5\left(x-1\right)}\)
  3. 3Cancel whole factors Cancel a complete common factor only where it is nonzero, and retain the original excluded inputs. Separate addends cannot be cancelled \(\displaystyle \frac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+5\right)}=\frac{x+3}{x+5},\quad x\ne-2,-5\)
  4. 4Use a reciprocal for division Reverse the divisor before multiplying. Both fractions must be defined, and the divisor must be nonzero \(\displaystyle \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c},\quad b,c,d\ne0\)
  5. 5Use a common denominator For addition or subtraction, form one denominator before combining numerators. Both original denominators must be nonzero \(\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd},\quad b,d\ne0\)
Year 12, VCE Units 3 & 4, Chapter 9, 4 difficulty levels, No calculatorBuild a worksheet

Algebraic fraction operations

Record the original denominator restrictions and exclude zeros of any divisor. Factorise before simplifying, use a reciprocal for division and a common denominator for addition or subtraction

\[\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c},\quad b,c,d\ne0\]
  1. Record the original restrictions Every original denominator must be nonzero. When dividing, the divisor must also be nonzero. Cancelling a factor does not restore an excluded input
  2. Factorise first Write each numerator and denominator as products before looking for common factors \(\displaystyle \frac{6x\left(x+2\right)}{5\left(x-1\right)}\)
  3. Cancel whole factors Cancel a complete common factor only where it is nonzero, and retain the original excluded inputs. Separate addends cannot be cancelled \(\displaystyle \frac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+5\right)}=\frac{x+3}{x+5},\quad x\ne-2,-5\)
  4. Use a reciprocal for division Reverse the divisor before multiplying. Both fractions must be defined, and the divisor must be nonzero \(\displaystyle \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c},\quad b,c,d\ne0\)
  5. Use a common denominator For addition or subtraction, form one denominator before combining numerators. Both original denominators must be nonzero \(\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd},\quad b,d\ne0\)

Worked example

Simplify the algebraic fraction and state all excluded values

\[\frac{3\left(a+2\right)}{2\left(a-4\right)}\div\frac{6\left(a-4\right)}{5a}\]
  1. \(\displaystyle \frac{3\left(a+2\right)}{2\left(a-4\right)}\times\frac{5a}{6\left(a-4\right)}\)Change division to multiplication by reversing the divisor
  2. \(\displaystyle \frac{3\times5}{2\times6}=\frac{5}{4}\)Reduce the numerical coefficient
  3. \(\displaystyle \frac{5a\left(a+2\right)}{4\left(a-4\right)^{2}}\)Combine the two matching factors in the denominator
  4. \(\displaystyle \frac{5a\left(a+2\right)}{4\left(a-4\right)^{2}},\quad a\ne0,\ a\ne4\)Keep the restrictions from the original denominators, so a cannot be 0 or 4.

Common mistakes

Cancelling terms
✕\(\displaystyle \frac{x+2}{x+5}=\frac{2}{5}\)✓\(\displaystyle \frac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)\left(x+5\right)}=\frac{x+3}{x+5},\quad x\ne-2,-5\)
Addends are not complete common factors, so they cannot be cancelled across a fraction bar
Using the divisor without a reciprocal
✕\(\displaystyle \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{c}{d}\)✓\(\displaystyle \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c},\quad b,c,d\ne0\)
Division by a fraction is multiplication by its reciprocal, so the divisor's numerator and denominator switch places
Losing excluded values
✕\(\displaystyle \frac{x\left(x+2\right)}{x}=x+2\)✓\(\displaystyle \frac{x\left(x+2\right)}{x}=x+2,\quad x\ne0\)
Cancelling the factor does not make the original denominator defined at zero, so the restriction must remain

Difficulty levels

introThis tier simplifies factored fractions and multiplies simple algebraic fractions
standardThis tier also divides fractions and combines fractions with related denominators
challengeThis tier combines several operations, subtraction and domain restrictions
examThis tier asks for equivalent forms and restrictions across mixed fraction operations

Questions about this topic

Why must the divisor be nonzero?+

Dividing by zero is undefined even when every denominator is nonzero. For an initial denominator of x minus one and a divisor with numerator x and denominator x plus two, exclude 1, −2 and 0

Why must factors be complete before they are cancelled?+

Cancellation divides a numerator and denominator by the same factor. Separate addends are not factors of the whole expression

How is a common denominator found for algebraic fractions?+

Factor the denominators and use each distinct factor with its greatest required power

Why are excluded values recorded after simplification?+

The simplified expression may hide a zero denominator or a zero divisor from the original expression. Keep both kinds of exclusion

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 9: Logarithmic calculus

In this chapter

Differentiate natural-logarithmic functions9A Logarithmic differentiation: Evaluate and determine function rules9A Logarithmic differentiation: Find the maximal domain of an algebraic function9A Logarithmic differentiation: Simplify and combine algebraic fractions9A Logarithmic differentiation: Expand and factor polynomial expressions9A Logarithmic differentiation: Evaluate a polynomial or recover a coefficient parameterApply logarithmic derivatives9B Applications of logarithmic derivatives: Form and analyse an inverse relation or functionFind the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular line9B Applications of logarithmic derivatives: Solve a linear equation9B Applications of logarithmic derivatives: Evaluate a polynomial or recover a coefficient parameter9B Applications of logarithmic derivatives: Solve a higher-degree polynomial equation or inequality9B Applications of logarithmic derivatives: Find the maximal domain of an algebraic function9B Applications of logarithmic derivatives: Evaluate and determine function rulesAnti-differentiate reciprocal and linear-reciprocal functions9C Reciprocal and linear-reciprocal antiderivatives: Simplify and combine algebraic fractions9C Reciprocal and linear-reciprocal antiderivatives: Expand and factor polynomial expressions9C Reciprocal and linear-reciprocal antiderivatives: Solve a linear equationIntegration by recognition with logarithmic components9D Recognising logarithmic derivatives: Expand and factor polynomial expressions9D Recognising logarithmic derivatives: Round to decimal places and significant figuresFind areas involving reciprocal curvesFind the area under a logarithmic curve using its inverseLogarithmic models in context9E Areas, inverse functions and logarithmic models: Form and analyse an inverse relation or function9E Areas, inverse functions and logarithmic models: Solve a linear equation9E Areas, inverse functions and logarithmic models: Solve a higher-degree polynomial equation or inequalityPerimeter and area of standard shapes9E Areas, inverse functions and logarithmic models: Round to decimal places and significant figures