Question

Form a polynomial whose zeroes are α² and β², where α and β are zeroes of the polynomial `p(x) = x^2 - 3sqrt(2)x + 4`.

Answer 1

1. Identify given zeroes

For the given polynomial `p(x) = x^2 - 3sqrt(2)x + 4`, we compare it with the general form ax² + bx + c.

Here:

a = 1

b = `-3sqrt(2)`

c = 4

Using the relationships between zeroes (α, β) and coefficients:

Sum of zeroes: `α + β = - b/a = 3sqrt(2)`

Product of zeroes: `αβ = c/a = 4` 

2. Calculate new sum

The zeroes of the new polynomial are α² and β². 

The sum of these new zeroes is:

α² + β² = (α + β)² – 2αβ

Substituting the known values:

`α^2 + β^2 = (3sqrt(2))^2 - 2(4)`

α² + β² = (9 × 2) – 8

= 18 – 8

= 0

3. Calculate new product

The product of the new zeroes is:

α² · β² = (αβ)²

Substituting the product αβ = 4:

α²β² = (4)²

= 16

4. Construct the polynomial

A polynomial with sum of zeroes S and product P is given by k(x² – Sx + P).

Taking k = 1:

x² – (10)x + 16

The polynomial whose zeroes are α² and β² is x² – 10x + 16.