SOLUTION: 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, Descart

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Question 33788This question is from textbook College Algebra
: 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
Thank You very much! This question is from textbook College Algebra

Answer by venugopalramana(3286) About Me  (Show Source):
You can put this solution on YOUR website!
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+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-5+%2B-+sqrt%28+5%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29+
x+=+%28-5+%2B-+sqrt%28+17+%29%29%2F%282%29+.LET US CLL THEM A AND B..THEN
P(X)=(X+2)(2X+1)(X-A)(X-B)