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; $f ( x ) = \left( x ^ { 2 } + 1 \right) \left( 4 x ^ { 2 } + 1 \right)$
C) $- 1$, multiplicity $2 ; f ( x ) = ( x + 1 ) ^ { 2 } \left( 4 x ^ { 2 } + 1 \right)$
D) 1 , multiplicity 2; $f ( x ) = ( x - 1 ) ^ { 2 } \left( 4 x ^ { 2 } + 1 \right)$

Question

Quizplus · Accepted Answer

Descartes' Rule of Signs indicates 2 or 0 positive real zeros and 0 negative real zeros. The Rational Zeros Theorem suggests possible rational zeros, but testing or further analysis shows 1 is a zero with multiplicity 2, leading to the factorization in option D.

Use Descartes' Rule of Signs and the Rational Zeros Theorem f(x)=4x4−8x3+5x2−2x+1f ( x ) = 4 x ^ { 4 } - 8 x ^ { 3 } + 5 x ^ { 2 } - 2 x + 1f(x)=4x4−8x3+5x2−2x+1

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