Question

Find all the zeros of the polynomial x³ + 3x² – 2x – 6, if two of its zeros are `-sqrt(2)` and `sqrt(2)`.

Answer 1

We know that if x = a is a zero of a polynomial, then x – a is a factor of f(x).

Since, `sqrt2` and `-sqrt2` are zeros of f(x).

Therefore

`(x+sqrt2)(x-sqrt2)=x^2-(sqrt2)^2`

= x² – 2

x² – 2 is a factor of f(x). Now, we divide x³ + 3x² – 2x – 6 by g(x) = x² – 2 to find the zero of f(x).

                        x + 3
`x^2 - 2")"overline(+ \cancel(x^3) + 3x^2 - \cancel(2x) - 6)`
            `+ \cancel(x^3) - 0     - \cancel(2x)`
            –                    +      
                    `+ \cancel(3x^2)  - \cancel(6)`
                    `+ \cancel(3x^2)  - \cancel(6)`
                     –           +      
                                0

By using division algorithm we have

f(x) = g(x) × q(x) – r(x)

x³ + 3x² – 2x – 6 = (x² – 2)(x + 3) – 0

x³ + 3x² − 2x − 6 `=(x+sqrt2)(x-sqrt2)(x+3)`

Hence, the zeros of the given polynomials are `-sqrt2, +sqrt2` and –3.