Use Descartes' Rule of Signs and the Rational Zeros Theorem $$f ( x ) = 4 x ^ { 4 } - 8 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$$

Multiple Choice

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the
zeros to factor f over the real numbers.
- $$f ( x ) = 4 x ^ { 4 } - 8 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1$$

A) $- 1,1 ; f ( x ) = ( x - 1 ) ( x + 1 ) \left( 4 x ^ { 2 } + 1 \right)$
B) no real roots; $\mathrm { f } ( \mathrm { x } ) = \left( \mathrm { x } ^ { 2 } + 1 \right) \left( 4 \mathrm { x } ^ { 2 } + 1 \right)$
C) $- 1$ , multiplicity 2; $f ( x ) = ( x + 1 ) ^ { 2 } \left( 4 x ^ { 2 } + 1 \right)$
D) 1 , multiplicity 2; $\mathrm { f } ( \mathrm { x } ) = ( \mathrm { x } - 1 ) ^ { 2 } \left( 4 \mathrm { x } ^ { 2 } + 1 \right)$