Seneca
Sign up

Polynomial Identities

Maths: Edexcel A Level Pure Maths, topic 1.3.4

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.

Example

Example

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

Expand the left-hand side using FOIL

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

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.
Recombine

Recombine

  • The first, third and fourth term of the expression are the same as the expansion of (x2−y2)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
Full proof

Full proof

Test yourself on Polynomial Identities

Practise this topic for free