Question

factorize x⁴-55x²+9. ​

Answer 1

Given, x⁴ + 5x² + 9 to factorise

= x⁴ + 6x² + 9 - x²

= [(x²)² + 2×3×(x²) + 3²] - (x)²

= (x² + 3)² - x²

It is in the form of a²-b² which can be written as (a+b)(a-b)

= (x²+3-x)(x²+3+x)

∴ x⁴+5x²+9 = (x²+3-x)(x²+3+x)

Answer 2

Rational Range

Let's assume this polynomial is obtained by difference of complete squares.

$x^4 - 55x^2 + 9$

$= ( x^4 - 6x^2 + 9 ) - 49x^2$

$= (x^2-3)^2 - (7x)^2$

$= ( x^2 + 7x - 3 )( x^2 - 7x - 3 )$

Real Range

Or we can solve for the two factors and fully factorize over reals. It is solved by factor theorem. By factor theorem, if α was our solution, x-α is necessarily a factor.

$x^2+7x-3 \implies \alpha ,\beta =\dfrac{-7 \pm \sqrt{61} }{2} \implies (x-\alpha )(x-\beta )$

$x^2-7x-3 \implies \gamma, \delta =\dfrac{7 \pm \sqrt{61} }{2} \implies (x-\gamma )(x-\delta )$

$\therefore (x+\dfrac{7-\sqrt{61} }{2} )(x+\dfrac{7+\sqrt{61} }{2} )(x-\dfrac{7-\sqrt{61} }{2} )(x-\dfrac{7+\sqrt{61} }{2} )$