Question 20643: Find the roots of this polynomial equation:
x³ - 5x² + 2x + 8 = 0
how do I do this i am so confused please help me. I need to know the steps to finding the answer..
Becca
Answer by AnlytcPhil(1806)   (Show Source):
You can put this solution on YOUR website! find the roots of each polynomial equation:
x(cubed)-5x(squared)+2x+8=0
`
x³ - 5x² + 2x + 8 = 0
`
Since the coefficient of x³ is 1, if it has any real rational roots,
they have to be a divisor of the absolute value of the constant term in
absolute value. So we can try +1, -1, +2, -2, +4, -4, +8, or -8
`
Try the easiest one first. Substitute x=1, you get
`
(1)³ - 5(1)² + 2(1) + 8 = 0
` ` ` ` `1 - 5 + 2 + 8 = 0
` ` ` ` ` ` ` ` ` ` `6 = 0
`
Nope, 1 is not a root. So try substituting x=-1, you get
`
(-1)³ - 5(-1)² + 2(-1) + 8 = 0
` ` ` ` -1 - 5 - 2 + 8 = 0
` ` ` ` ` ` ` ` ` ` `0 = 0
`
Yes that's a solution, so if x = -1 is a root, that means
`
x + 1 is a factor, because if you set that = 0 you'll get x = -1.
`
So divide by x+1, you use synthetic division, using -1 in the far left:
`
-1 | 1 -5 2 `8
` `|___-1_6_-8_
` ` `1 -6 8 `0
`
The numbers on the bottom mean x²-6x+8 with the last number 0
being the remainder, which is how you know x+1 was a factor of
the polynomial. So you now have the polynomial equation factored
as
`
(x+1)(x²-6x+8) = 0
`
You can now factor the trinomial in the second parentheses as
(x-2)(x-4) and get
`
(x+1)(x-2)(x-4)=0
`
Set each of those three parenthetical expressions = 0, and get -1, 2,
and 4.
`
Edwin
AnlytcPhil@aol.com
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