SOLUTION: Please help me solve this equation! List the rational roots and determine them:{{{2x^3-7x+2=0}}}

Question 166084: Please help me solve this equation! List the rational roots and determine them:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients

So let's list the factors of 2 (the last coefficient):

Now let's list the factors of 2 (the first coefficient):

Now let's divide each factor of the last coefficient by each factor of the first coefficient

Now simplify

These are all the distinct rational zeros of the function that could occur

=================================================================

Now let's test all of the possible zeros:

Let's see if the possible zero is really a root for the function

So let's make the synthetic division table for the function given the possible zero :

1	\|	2	0	-7	2
	\|		2	2	-5
		2	2	-5	-3

Since the remainder (the right most entry in the last row) is NOT equal to zero, this means that is NOT a zero of

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Let's see if the possible zero is really a root for the function

So let's make the synthetic division table for the function given the possible zero :

1/2	\|	2	0	-7	2
	\|		1	1/2	-13/4
		2	1	-13/2	-5/4

2	\|	2	0	-7	2
	\|		4	8	2
		2	4	1	4

-1	\|	2	0	-7	2
	\|		-2	2	5
		2	-2	-5	7

-1/2	\|	2	0	-7	2
	\|		-1	1/2	13/4
		2	-1	-13/2	21/4

-2	\|	2	0	-7	2
	\|		-4	8	-2
		2	-4	1	0

Since the remainder (the right most entry in the last row) is equal to zero, this means that is a zero of

Also, it turns out that the numbers 2,-4, and 1 (the bottom row of numbers) form the coefficients of the quotient. So this means that

Now multiply both sides by

So this means that factors to

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Now let's solve

Notice we have a quadratic equation in the form of where , , and

Let's use the quadratic formula to solve for x

Start with the quadratic formula

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Subtract from to get

Multiply and to get .

Simplify the square root (note: If you need help with simplifying square roots, check out this solver)

or Break up the expression.

or Reduce

So the last two zeros are or

which approximate to or

=============================================================

Answer:

So after all of that, we get the three zeros: , or

Which in decimal form are , , or
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