1.3.3

Proving an Identity

Test yourself

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'.

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Identity

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

  • Consider the left-hand side of the identity (n+3)(n+2)3n+2(n+3)(n+2)-3n+2.
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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+63n+2(n+3)(n+2)-3n+2 = n^2 + 2n + 3n + 6 - 3n+2
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Collect terms

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

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

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

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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