1.3.4

Polynomial Identities

Test yourself

Polynomial Identities

An identity equation is true for all values of the variable. To prove an identity holds, show that the left-hand side equals the right-hand side.

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Example

  • Show that (x2+y2)2=(x2y2)2+(2xy)2(x^2 + y^2)^2 = (x^2-y^2)^2+(2xy)^2
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Expand the left-hand side using FOIL

  • Collect any like terms together.
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Add expression equivalent to zero

  • Adding (2x2y2+2x2y2)(-2x^2y^2+2x^2y^2) does not change the expression as it is equal to zero.
  • It allows us to combine the other terms in a way that resembles the right-hand side of the equation.
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Recombine

  • The first, third and fourth term of the expression are the same as the expansion of (x2y2)2(x^2-y^2)^2.
  • By substituting different values of xx and yy, we can use this identity to generate Pythagorean triples.
    • 32 + 42 = 52

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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