Find all zeros of the function and write the polynomial as a product of linear factors.
-$$f ( x ) = x ^ { 3 } - x ^ { 2 } + 16 x - 16$$
A) $f ( x ) = ( x - 1 ) ( x + 4 i ) ( x - 4 i )$
B) $f ( x ) = ( x - 1 ) ( x + 1 ) ( x + 16 )$
C) $f ( x ) = ( x - 25 ) ( x + i ) ( x - i )$
D) $f ( x ) = ( x - 1 ) ( x + 4 ) ( x - 4 )$

Question

Quizplus · Accepted Answer

To find the zeros of the function, we can factor the given polynomial. Factoring by grouping, we get $x^2(x - 1) + 16(x - 1)$, which simplifies to $(x - 1)(x^2 + 16)$. The quadratic factor $x^2 + 16$ can be further factored into $(x + 4i)(x - 4i)$ using the complex numbers, since it has no real roots. Thus, the polynomial can be written as a product of linear factors as $f(x) = (x - 1)(x + 4i)(x - 4i)$.

Find All Zeros of the Function and Write the Polynomial f(x)=x3−x2+16x−16f ( x ) = x ^ { 3 } - x ^ { 2 } + 16 x - 16f(x)=x3−x2+16x−16

Find All Zeros of the Function and Write the Polynomial $f ( x ) = x ^ { 3 } - x ^ { 2 } + 16 x - 16$