Write the polynomial $x ^ { 4 } - 7 x ^ { 2 } - 8$ as the product of factors that are irreducible over the rationals.
A) $\left( x ^ { 2 } - 8 \right) \left( x ^ { 2 } - 1 \right)$
B) $\left( x ^ { 2 } - 8 \right) \left( x ^ { 2 } + 1 \right)$
C) $\left( x ^ { 2 } + 8 \right) \left( x ^ { 2 } + 1 \right)$
D) $\left( x ^ { 2 } + 8 \right) \left( x ^ { 2 } - 1 \right)$
E) $\left( x ^ { 2 } - 7 \right) \left( x ^ { 2 } - 1 \right)$

Question

Quizplus · Accepted Answer

The polynomial $x^4 - 7x^2 - 8$ can be factored as $(x^2 - 8)(x^2 + 1)$. Both factors are irreducible over the rationals because $x^2 - 8$ cannot be factored further without using irrational numbers (roots of 8 are irrational), and $x^2 + 1$ has no real roots (its roots are imaginary).

Write the Polynomial x4−7x2−8x ^ { 4 } - 7 x ^ { 2 } - 8x4−7x2−8

Write the Polynomial $x ^ { 4 } - 7 x ^ { 2 } - 8$