Find all zeros of the function and write the polynomial as a product of linear factors.
-$$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + 8 x + 8$$
A) $f ( x ) = ( x + 2 ) ( x + 1 + i \sqrt { 3 } ) ( x - 1 - i \sqrt { 3 } )$
B) $f ( x ) = ( x - 2 ) ( x + 3 + 1 i ) ( x - 3 - 1 i )$
C) $f ( x ) = ( x + 2 ) ( x + 1 + i \sqrt { 3 } ) ( x + 1 - i \sqrt { 3 } )$
D) $f ( x ) = ( x - 2 ) ( x + 3 + 1 i ) ( x + 3 - 1 i )$

Question

Quizplus · Accepted Answer

The polynomial can be factored by grouping or using the rational root theorem to find a real root, which is $x = -2$. After dividing the polynomial by $x + 2$, we get a quadratic equation whose roots can be found using the quadratic formula, leading to complex roots $x = -1 \pm i\sqrt{3}$. Thus, the polynomial can be written as $f(x) = (x + 2)(x + 1 + i\sqrt{3})(x + 1 - i\sqrt{3})$.

Find All Zeros of the Function and Write the Polynomial f(x)=x3+4x2+8x+8f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + 8 x + 8f(x)=x3+4x2+8x+8

Find All Zeros of the Function and Write the Polynomial $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + 8 x + 8$