Answer to Question #145470 in Algebra for Rolando

Question

LET’S TRY THIS!
List all possible rational roots of each polynomial equation.

1. 3x² + 2x – 1=0
Ans.

2. x³-27
Ans.

State the possible rational zeros for each polynomial. Then find all rational zeros.
3. P(x) x³ +4x² + 5x + 2
Possible zeros:
Rational zeros:

4. 3x³ + 11x² + 5x – 3
Possible zeros:
Rational zeros:

Answer 1

1.

$a_0=1,\:\quad a_n=3$

$\mathrm{The\:divisors\:of\:}a_0:\quad 1,\:\quad \mathrm{The\:divisors\:of\:}a_n:\quad 1,\:3$

$\mathrm{The\:following\:rational\:numbers\:are\:possible \,candidates\:for \,roots:}\quad \pm \frac{1}{1}, \pm \frac{1}{3}$

2.

$a_0=27,\:\quad a_n=1$

$\mathrm{The\:divisors\:of\:}a_0:\quad 1,\:3,\:9,\:27,\:\quad \mathrm{The\:divisors\:of\:}a_n:\quad 1$

$\mathrm{The\:following\:rational\:numbers\:are\:possible \,candidates\:of \,roots:}\quad \\ \pm \frac{1}{1}, \pm\frac{3}{1}, \pm \frac{9}{1}, \pm \frac{27}{1}$

3.

$a_0=2,\:\quad a_n=1$

$\mathrm{The\:divisors\:of\:}a_0:\quad 1,\:2,\:\quad \mathrm{The\:divisors\:of\:}a_n:\quad 1$

$\mathrm{The\:following\:rational\:numbers\:are\:possible \,candidates\:of \,roots:}\quad \pm \frac{1}{1}, \pm \frac{2}{1}$

$f(-1)=-1+4-5+2=0$

$f(-2)=-8+16-10+2=0$

$f(1)=1+4+5+2=12$

$f(2)=46$

$x=-1,x=-2$

4.

$a_0=3,\:\quad a_n=3$

$\mathrm{The\:divisors\:of\:}a_0:\quad 1,\:3,\:\quad \mathrm{The\:divisors\:of\:}a_n:\quad 1,\:3$

$\mathrm{The\:following\:rational\:numbers\:are\:possible \,candidates\:of \,roots:}\quad \pm \frac{1}{1}, \pm \frac{3}{1}, \pm \frac{1}{3}$

$f(-1)=0$

$f(1/3)=0$

$f(-3)=0$

$x=-3, x=-1, x=1/3$