Find all the zeroes of the polynomial 3x⁴ + 6x³ – 2x² – 10x – 5 if two of its zeroes are and .

We have the polynomial,

3x⁴ + 6x³ – 2x² – 10x – 5

And two of its zeroes are and -.

If x = is a zero, then (x -) is a factor of the polynomial.

And if x = is another zero, then (x +) is the other factor of the polynomial.

⇒ (x -) (x +) is also a factor.

Or (x² – 5/3) is also a factor.

If 3x⁴ + 6x³ – 2x² – 10x – 5 is divided by (x² – 5/3), we can find other factors also.

So we have got 3x² + 6x + 3, now by factorizing it we can obtain other factors as well.

Splitting 3x² + 6x + 3,

3x² + 6x + 3 = 3x² + 3x + 3x + 3 [∵ 6x is split into (3x + 3x) in such a way that (3x × 3x) = 9 and (3x + 3x) = 6x]

= 3x (x + 1) + 3 (x + 1)

= (3x + 3) (x + 1) [By taking common]

= 3 (x + 1) (x + 1)

= 3 (x + 1)²

Taking 3(x + 1)² = 0, we get

(x + 1)² = 0

⇒ x + 1 = 0

⇒ x = -1

Thus, all zeroes of the polynomial 3x⁴ + 6x³ – 2x² – 10x – 5 are and -1.