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Mathematics 2.1 Manipulate algebraic expressions and solve equations

Factorising algebraic expressions

Factorise is the reverse of expand i.e. put in brackets.

To factorise, rewrite the expression as factors multiplied together.

Two terms

Look for the largest number that will divide into all terms and any common letters.

 e.g.   12x + 20xy = 4x(3 + 5y)                  
          This is because 4x is the largest factor of both 12x and
          20xy. 

Check by expanding your answer.

See online examples of common factors.

Two terms: Difference of two squares

Look for two square terms that have been subtracted.

 e.g.   4x2 – 9     
          = (2x)2 – 32         
          = (2x + 3)(2x – 3) 

See online examples of factoring the difference of two squares and practice with factoring difference of two squares.

Three terms

Look for a common factor first and then factorise further.

 e.g.   3x2 – 6x – 15 = 3(x2 – 2x – 5) 

If no common factor, look for two numbers that multiply to give the number by itself (constant term) and add to give the number in front of x.

 e.g.   x2 – 2x - 15 = (x + 3)(x – 5)          
         This is because –5 times 3 = –15 and –5 plus 3 = –2 

Check

 Hence   3x2 – 6x – 15 = 3(x + 3)(x – 5) 

If there is no number written in front of x remember it is really a 1.

 e.g.   x2 + x – 6 = (x + 3)(x – 2)          
         This is because 3 times -2 = –6 and 3 plus –2 = 1 

See notes on:

• Factorising quadratic expressions
• Factoring quadratic expressions

Harder factorising

If there is a number in front of x² that is not a common factor use guess and check or the following method.

Two numbers that multiply to give –6 and add to give +5 are +6 and –1. Write 5x as 6x - x.

 e.g.   2x2 + 5x – 3= 2x2 + 6x – x – 3 

When there are four terms, factorise in pairs first.

 e.g.   2x2 + 6x – x – 3 = 2x(x + 3) – 1(x + 3) 

Take out the common bracket (x + 3).

 e.g.   2x(x + 3) – 1(x + 3) = (x + 3)(2x – 1) 

See notes on:

• Factoring
• Algebasics – Section 8: Factoring (Macromedia Flash® and sound required)

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