Question

X4-7x2+1 is reducible or irreducible over z?

Answer 1

Factorizing for integers

After applying complete the square method

$\sf{\to (x^4+2x^2+1)-9x^2}$

$\sf{\to (x^2+1)^2-(3x)^2}$

After this identity is applied.

$\sf{\to (x^2+3x+1)(x^2-3x+1)$

Therefore, the polynomial is reducible for integers.

Further factorized

Using: Factor Theorem

Concept: A polynomial having α as root has a factor (x-α).

Application:-

Two quadratic factors have roots at

$\sf{(x^2+3x+1)=0 \: or \:(x^2-3x+1)=0}$

$\sf{\to (x+\dfrac{-3+\sqrt{5} }{2} ) (x+\dfrac{-3-\sqrt{5} }{2} ) (x+\dfrac{3+\sqrt{5} }{2} ) (x+\dfrac{3-\sqrt{5} }{2} )$

Since all the factors are linear, it is fully factorized.

Hence, the polynomial cannot be factorized further.

Therefore, reducible for real numbers.

Answer 2

Using: Factor Theorem

Concept: A polynomial having α as root has a factor (x-α).

Application:-