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Proving an Identity

Maths: Edexcel A Level Pure Maths, topic 1.3.3

How do you prove an identity?

To prove an identity is always true, you need to show one side is the same as the other. The identity symbol is ≡, which means 'equivalent'.

Identity

Identity

  • Prove that (n+3)(n+2)−3n+2≡n2+2n+8(n+3)(n+2)-3n+2\equiv n^2+2n+8.
Left-hand side

Left-hand side

  • Consider the left-hand side of the identity (n+3)(n+2)−3n+2(n+3)(n+2)-3n+2.
Expand brackets

Expand brackets

  • First, expand the brackets using FOIL. This step is known to always be true.
  • (n+3)(n+2)−3n+2=n2+2n+3n+6−3n+2(n+3)(n+2)-3n+2 = n^2 + 2n + 3n + 6 - 3n+2
Collect terms

Collect terms

  • Collect all the terms of equal powers of n to give:
  • n2+2n+8n^2 + 2n+8
Right-hand side

Right-hand side

  • Comparing to the right-hand side of the identity, we see that they are always equal.
Final statement

Final statement

  • The final statement is that the identity holds:
  • So (n+3)(n+2)−3n+2≡n2+2n+8(n+3)(n+2)-3n + 2\equiv n^2+2n+8
Whole proof

Whole proof

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