P is a polynomial such that Px2 1 2 x4 5 x2 6 Find P x2

P is a polynomial such that P(x² + 1) = - 2 x⁴ + 5 x² + 6.

Find P(- x² + 3)

Solution

P(x² + 1) = - 2 x⁴ + 5 x² + 6

Let t = x² + 1 which also gives x² = t - 1

Substitute x² by t - 1 in P to obtain: P(t) = - 2 (t - 1)² + 5 (t - 1) + 6 = -2 t ² + 9t - 1

Now let t = - x² + 3 and substitute in P(t) above to obtain

P(- x² + 3) = -2 (- x² + 3) ² + 9 (- x² + 3) - 1 = -2 x ⁴ + 3 x ² + 8