Factorise the polynomial x⁴ − 10x² + 22 into product of two quadratic polynomials. 

Given:

Polynomial: x⁴ − 10x² + 22

Formula used:

Quadratic formula: For an equation ax² + bx + c = 0, x = \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Calculations:

Let y = x². The given polynomial can be written as a quadratic equation in y:

y² - 10y + 22 = 0

Using the quadratic formula to find the roots for y, where a = 1, b = -10, c = 22:

y = \(\frac{-(-10) \pm \sqrt{(-10)^2 - 4 \times 1 \times 22}}{2 \times 1}\)

⇒ y = \(\frac{10 \pm \sqrt{100 - 88}}{2}\)

⇒ y = \(\frac{10 \pm \sqrt{12}}{2}\)

⇒ y = \(\frac{10 \pm 2\sqrt{3}}{2}\)

⇒ y = 5 \(\pm\) \(\sqrt{3}\)

So, the two roots for y are:

y₁ = 5 + \(\sqrt{3}\)

y₂ = 5 - \(\sqrt{3}\)

Therefore, the quadratic in y can be factored as:

(y - y₁)(y - y₂) = (y - (5 + \(\sqrt{3}\)))(y - (5 - \(\sqrt{3}\)))

Substitute back y = x²:

(x² - (5 + \(\sqrt{3}\)))(x² - (5 - \(\sqrt{3}\)))

∴ The factorization of the polynomial x⁴ − 10x² + 22 into a product of two quadratic polynomials is (x² - (5 + \(\sqrt{3}\)))(x² - (5 - \(\sqrt{3}\))).