Write the polynomial $x ^ { 4 } - 6 x ^ { 2 } - 72$ as the product of linear and quadratic factors that are irreducible over the reals.
A) $( x + 2 \sqrt { 3 } ) ( x + 2 \sqrt { 3 } ) \left( x ^ { 2 } + 6 \right)$
B) $( x - 2 \sqrt { 3 } ) ( x - 2 \sqrt { 3 } ) \left( x ^ { 2 } + 6 \right)$
C) $( x + 2 \sqrt { 3 } ) ( x - 2 \sqrt { 3 } ) \left( x ^ { 2 } - 6 \right)$
D) $( x + 2 \sqrt { 3 } ) ( x - 2 \sqrt { 3 } ) \left( x ^ { 2 } + 6 \right)$
E) $( x + 2 \sqrt { 3 } ) ( x + 2 \sqrt { 3 } ) \left( x ^ { 2 } - 6 \right)$

Question

Quizplus · Accepted Answer

The polynomial can be factored as $x^4 - 6x^2 - 72 = (x^2 - 6)^2 - 72 = (x^2 - 6 - 6\sqrt{3})(x^2 - 6 + 6\sqrt{3})$, which simplifies to $(x - 2\sqrt{3})(x + 2\sqrt{3})(x^2 + 6)$, matching choice D.

Write the Polynomial x4−6x2−72x ^ { 4 } - 6 x ^ { 2 } - 72x4−6x2−72

Write the Polynomial $x ^ { 4 } - 6 x ^ { 2 } - 72$