Find all the common zeroes of the polynomials: x³ + 5x² - 9x - 45 and x³ + 8x² + 15x.

Let P(x) = x ³ + 5x ² – 9x – 45 and Q (x) = x ³ + 8x ² + 15x.

Consider P(x) = x ³ + 5x ² – 9x – 45

= x ² (x + 5) – 9(x + 5)
= (x + 5) (x ² – 9)
= (x + 5) (x + 3) (x – 3)

(using identity: (a ² – b ² ) = (a + b)(a - b))

∴ On putting P(x) = 0, we get x = -5, x = -3, x = 3 as its zeroes.

Now consider Q (x) = x ³ + 8x ² + 15x

= x (x ² + 8x + 15)
= x (x ² + 5x + 3x + 15)

= x (x(x + 5)+ 3(x + 5))

= x (x + 5) (x + 3)
∴ On putting Q(x) = 0, we get x = 0, x = -5, x = -3 as its zeroes.

∴ The common zeroes of the polynomials are x = -5 and x = -3.