Question

Obtain all other zeroes of (x⁴ + 4x³ – 2x² – 20x – 15) if two of its zeroes are `sqrt(5)` and `-sqrt(5)`.

Answer 1

The given polynomial is f(x) = x⁴ + 4x³ – 2x² – 20x – 15.

Since `(x - sqrt(5))` and `(x + sqrt(5))` are the zeroes of f(x) it follows that each one of `(x - sqrt(5))` and `(x + sqrt(5))` is a factor of f(x).

Consequently, `(x - sqrt(5)) (x + sqrt(5)) = (x^2 - 5)` is a factor of f(x).

On dividing f(x) by (x² – 5), we get:

`x^2 - 5")"overline(x^4 + 4x^3 - 2x^2 - 20x - 15)"("2x^2 - 3x + 1`
             x⁴            – 5x²
             –                +                          
             4x³ + 3x² – 20x – 15
             4x³           – 20
             –               +                           
                       3x² – 15
                       3x² – 15
                    –        +                           
                            x                             
f(x) = 0

⇒ x⁴ + 4x³ – 7x² – 20x – 15 = 0

⇒ (x² – 5) (x² + 4x + 3) = 0

⇒ `(x - sqrt(5)) (x + sqrt(5)) (x + 1) (x + 3) = 0`

⇒ x = `sqrt(5)` or x = `-sqrt(5)` or x = –1 or x = –3

Hence, all the zeroes are `sqrt(5), -sqrt(5)`, –1 and –3.