Algebraic fractions

• What is an algebraic fraction?

  An algebraic fraction is a fraction with an algebraic expression on the top (numerator) and/or the bottom (denominator).
• How do you simplify an algebraic fraction?

  You simplify an algebraic fraction by factorising fully the top and bottom and then cancelling common factors.
• What should you do if you are asked to simplify an algebraic fraction and need to factorise?

  If you need to factorise, it is likely that one of the factors will be the same on the top and the bottom.

  
• What is the first step in simplifying the algebraic fraction $\frac{4x + 6}{2x^2 - 7x - 15}$?


  • Factorise the numerator: $2(2x + 3)$
  • Factorise the denominator using your preferred method.

  
• What is the lowest common denominator (LCD) of the fractions $x - 2$ and $x +$$5$?


  The LCD is $(x - 2)(x + 5)$.
• How do you find the LCD of $x$ and $2x$?


  The LCD is $2x$ since $2x$ already includes an $x$.
• What is the LCD of $x +$$2$ and $(x + 2)(x - 1)$?


  The LCD is $(x + 2)(x - 1)$ since it already includes $(x + 2)$.
• How do you combine two algebraic fractions?

  You write each fraction over the lowest common denominator and then add or subtract the numerators.
• What should you do after writing each fraction over the lowest common denominator?

  You should multiply the numerators of each fraction by the same amount as the denominators.
• What is the final step after combining the fractions?

  You check to see if the numerator can be factorised and cancelled.
• What are the steps to multiply algebraic fractions?

  1. Simplify both fractions by fully factorising and cancelling common factors.
  2. Multiply the numerators together.
  3. Multiply the denominators together.
  4. Check for any further factorising and cancelling.

  
• How do you divide algebraic fractions?

  1. Flip (reciprocate) the second fraction and replace ÷ with ×.
  2. Follow the same rules for multiplying two fractions.

  
• What is the first step in solving an equation that contains algebraic fractions?

  One method is to deal with the algebraic fractions by adding or subtracting them first and then solving the equation.

  
• What should you do to remove the fractions in an equation?

  You should multiply both sides by everything on the denominator.
• What is the second method for solving equations with algebraic fractions?

  The second method is to multiply everything in the fraction by each of the expressions on the denominator.

  
• What are the steps to solve the equation $\frac{4}{x - 3} +$$\frac{5}{x + 1} =$$5$?


  1. Multiply every term by $(x - 3)(x + 1)$.
  2. Expand the brackets on both sides and simplify.
  3. Rearrange the equation to solve.
  4. Solve the resulting equation.
• What should you do after expanding the brackets and simplifying?

  You should rearrange the equation so that it is in a form that can be solved.
• What is the final step in solving the equation $5x^2 - 19x - 4 =$$0$?


  The final step is to factorise the equation and find the values of $x$.
• How can you show that the equation $2p +$$3 - 5p =$$6p$ can be written as $6p^3 +$$18p^2 +$$3p +$$15 =$$0$?


  1. Clear the fractions by multiplying both sides by the denominators.
  2. Expand the brackets.
  3. Collect like terms.
  4. Rearrange to complete the equation.