Question

Find the zeroes of the quadratic polynomial `2x^2 - (1 + 2sqrt(2))x + sqrt(2)` and verify the relationship between the zeroes and coefficients of the polynomial.

Answer 1

Given polynomial: `2x^2 - (1 + 2sqrt(2))x + sqrt(2)`

So the coefficients are a = 2, b = `-(1 + 2sqrt(2))`,  c = `sqrt2`

Factorisation of the polynomial:

We split the middle term `-(1 + 2sqrt2)x` as `-x - 2sqrt2x`.

So the polynomial becomes:

`2x^2 - x - 2sqrt(2)x + sqrt(2)`

Taking common factors:

`(2x^2 - x) - (2sqrt2x - sqrt2)`

= `x(2x - 1) - sqrt2(2x - 1)`

= `(2x - 1)(x - sqrt(2))`

Finding the zeroes of the polynomial:

Set each factor equal to zero:

(i) 2x − 1 = 0

x = `1/2`

(ii) `x - sqrt2` = 0

x = `sqrt2`

Hence, the zeroes of the polynomial are `1/2` and `sqrt(2)`.

Verification of the relationship between zeroes and coefficients:

Let α = `1/2`, β = `sqrt2`

Sum of zeroes:

α + β

= `1/2 + sqrt2`

Using formula:

`(-b)/a`

= `(-(1 + 2sqrt(2)))/2`

= `(1 + 2 sqrt2)/2`

= `1/2 + sqrt2`

Product of zeroes:

α × β

= `1/2 xx sqrt2`

= `sqrt2/2`

Using formula:

`c/a`

= `sqrt(2)/2`

Since α + β = `(-b)/a` and α × β = `c/a` both conditions are satisfied.

Hence, the relationship between the zeroes and the coefficients is verified.