Write the polynomial as the product of linear and quadratic factors that are irreducible over the reals. ? F(x)= x⁴ + 42x² - 343 ? A) $\left( x ^ { 2 } + 49 \right) ( x + \sqrt { 7 } ) ( x - 7 )$ B) $\left( x ^ { 2 } + 49 \right) ( x + \sqrt { 7 } ) ( x + \sqrt { 7 } )$ C) $\left( x ^ { 2 } + 49 \right) ( x - \sqrt { 7 } ) ( x - \sqrt { 7 } )$ D) $\left( x ^ { 2 } + 49 \right) ( x + \sqrt { 7 } ) ( x - \sqrt { 7 } )$ E) $\left( x ^ { 2 } + 49 \right) ( x + 7 ) ( x - \sqrt { 7 } )$

Question

Quizplus · Accepted Answer

The polynomial $F(x) = x^4 + 42x^2 - 343$ can be factored into $(x^2 + 49)(x^2 - 7)$. The quadratic $x^2 + 49$ is irreducible over the reals because it has no real roots. The quadratic $x^2 - 7$ can be factored into $(x + \sqrt{7})(x - \sqrt{7})$, which are linear factors. Therefore, the correct factorization is $(x^2 + 49)(x + \sqrt{7})(x - \sqrt{7})$.

Write the Polynomial as the Product of Linear and Quadratic (x2+49)(x+7)(x−7)\left( x ^ { 2 } + 49 \right) ( x + \sqrt { 7 } ) ( x - 7 )(x2+49)(x+7​)(x−7)

Write the Polynomial as the Product of Linear and Quadratic $\left( x ^ { 2 } + 49 \right) ( x + \sqrt { 7 } ) ( x - 7 )$