A polynomial P is given 4 5x2 Px 36 a Factor P into linear a

A polynomial P is given 4 5x2 P(x) 36 (a) Factor P into linear and irreducible quadratic factors with real coefficients P(x) (b) Factor P completely into linear factors with complex coefficients P(x)

Solution

P(x) = x^4 +5x^2 -36

a) P(x) = ( x^4 +5x^2 -36

plug x^2 =y

y^2 +5y -36

Factorising in the same way as quadratic function:

y = (- 5+/- sqrt(25 + 144)/2

= ( -5 +/- 13)/2

= -9 , 4

So, x^4 +5x^2 -36 = (x^2 +9)(x^2 -4)

b) P (x) = (x^2 +9)(x^2 -4)

= (x +3i)(x-3i)(x+2)(x-2)