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.
-$$f ( x ) = 2 x ^ { 3 } + 13 x ^ { 2 } + 10 x + 25$$
A) $- 1 , \frac { 5 } { 2 } , 5 ; f ( x ) = ( 2 x - 5 ) ( x - 5 ) ( x + 1 )$
B) $- 5 , \frac { 5 } { 2 } , 1 ; f ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x + 5 )$
C) $1 , \frac { 2 } { 5 } , - 5 ; f ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x + 5 )$
D) $- 1 , \frac { 2 } { 5 } , - 5 ; f ( x ) = ( 2 x - 5 ) ( x - 5 ) ( x + 1 )$

Question

Quizplus · Accepted Answer

Descartes' Rule of Signs indicates there can be 3 or 1 positive real zeros (changes in sign in the original polynomial) and 1 negative real zero (one change in sign in the polynomial with $x$ replaced by $-x$). The Rational Zeros Theorem suggests possible rational zeros as factors of $25/2$, which includes $\pm1$, $\pm5$, $\pm5/2$, etc. Testing or directly checking the given options, option A correctly identifies the zeros and their corresponding factorization.

Use Descartes' Rule of Signs and the Rational Zeros Theorem f(x)=2x3+13x2+10x+25f ( x ) = 2 x ^ { 3 } + 13 x ^ { 2 } + 10 x + 25f(x)=2x3+13x2+10x+25

Use Descartes' Rule of Signs and the Rational Zeros Theorem $f ( x ) = 2 x ^ { 3 } + 13 x ^ { 2 } + 10 x + 25$