SOLUTION: (22 pts) Consider the polynomial f(x) = 2x^3 – 3x^2 – 8x – 3. (i) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.

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i)

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 -3 (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







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ii)

With the help of a graphing calculator, we see that -1 is a zero of

note: let me know if you need to find the zeros a different way.

So let's set up a synthetic division table by placing the value -1 in the upper left corner and placing the coefficients of the polynomial to the right of -1.

-1	|	2	-3	-8	-3
	|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 2)

-1	|	2	-3	-8	-3
	|
		2

Multiply -1 by 2 and place the product (which is -2) right underneath the second coefficient (which is -3)

-1	|	2	-3	-8	-3
	|		-2
		2

Add -2 and -3 to get -5. Place the sum right underneath -2.

-1	|	2	-3	-8	-3
	|		-2
		2	-5

Multiply -1 by -5 and place the product (which is 5) right underneath the third coefficient (which is -8)

-1	|	2	-3	-8	-3
	|		-2	5
		2	-5

Add 5 and -8 to get -3. Place the sum right underneath 5.

-1	|	2	-3	-8	-3
	|		-2	5
		2	-5	-3

Multiply -1 by -3 and place the product (which is 3) right underneath the fourth coefficient (which is -3)

-1	|	2	-3	-8	-3
	|		-2	5	3
		2	-5	-3

Add 3 and -3 to get 0. Place the sum right underneath 3.

-1	|	2	-3	-8	-3
	|		-2	5	3
		2	-5	-3	0

Since the last column adds to zero, this means that -1 is a zero of (this confirms our original claim).



Now lets look at the bottom row of coefficients:

The first 3 coefficients (2,-5,-3) form the quotient




So


Basically factors to


Now lets find the zeros for .


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


Rewrite as


Add to to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the zeros of are , , or