Question 33788
I'm trying to find all rational zeros of the following polynomial and then find the irrational zeros. using the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques:
P(x) = 2x^4 + 15x^3 +31 x^2 + 20x + 4
THERE ARE NO CHANGE OF SIGNS.HENCE THERE ARE NO POSITIVE REAL ROOTS
P(-X)=2X^4-15X^3+31X^2-20X+4
THERE ARE 2 CHANE OF SIGNS.HENCE THERE COULD BE MAIMUM 2 NEGATIVE REAL ROOTS.
NOW POSIBLE ROOTS ARE FACTORS OF CONSTAN TERM /FACTORS OF X^4..THAT IS...
-1/1,-2/1,-4/1,-1/2
WE FIND THAT P(-2)=0 AND P(-0.5)=0...HENCE X+2 IS A FACTOR AND X+0.5 IS A FACTOR..DIVIDING WE GET

-2|.......2.......15.........31.........20..........4
..|.......0.......-4........-22........-18.........-4
---------------------------------------------------------
-0.5|.....2.......11.........9..........2...........0 
....|.....0.......-1........-5.........-2
-------------------------------------------------------------
..........2.......10.........4..........0
HENCE QUOTIENT IS...
2X^2+10X+4...OR...
P(X)=(X+2)(X+0.5)2(X^2+5X+2)=(X+2)(2X+1)(X^2+5X+2)
ROOTS OF X^2+5X+2=0....ARE GOT USING QUADRATIC FORMLA
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
{{{x = (-5 +- sqrt( 5^2-4*1*2 ))/(2*1) }}}
{{{x = (-5 +- sqrt( 17 ))/(2) }}}.LET  US CLL THEM A AND B..THEN 
P(X)=(X+2)(2X+1)(X-A)(X-B)