SOLUTION: 1. Given the polynomial f(x) = 4x^4 + 5x^3 + 7x^2 - 34x + 8 (a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.

Question 149741: 1. Given the polynomial f(x) = 4x^4 + 5x^3 + 7x^2 - 34x + 8

(a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.

(b) Find all of the zeros of the given polynomial. Show the procedure

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)

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 8 (the last coefficient):

Now let's list the factors of 4 (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

b)

Now let's use synthetic division to test each possible zero:

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

1/2	\|	4	5	7	-34	8
	\|		2	7/2	21/4	-115/8
		4	7	21/2	-115/4	-51/8

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 make the synthetic division table for the function given the possible zero :

1/4	\|	4	5	7	-34	8
	\|		1	3/2	17/8	-255/32
		4	6	17/2	-255/8	1/32

2	\|	4	5	7	-34	8
	\|		8	26	66	64
		4	13	33	32	72

4	\|	4	5	7	-34	8
	\|		16	84	364	1320
		4	21	91	330	1328

8	\|	4	5	7	-34	8
	\|		32	296	2424	19120
		4	37	303	2390	19128

-1	\|	4	5	7	-34	8
	\|		-4	-1	-6	40
		4	1	6	-40	48

-1/2	\|	4	5	7	-34	8
	\|		-2	-3/2	-11/4	147/8
		4	3	11/2	-147/4	211/8

-1/4	\|	4	5	7	-34	8
	\|		-1	-1	-3/2	71/8
		4	4	6	-71/2	135/8

-2	\|	4	5	7	-34	8
	\|		-8	6	-26	120
		4	-3	13	-60	128

-4	\|	4	5	7	-34	8
	\|		-16	44	-204	952
		4	-11	51	-238	960

-8	\|	4	5	7	-34	8
	\|		-32	216	-1784	14544
		4	-27	223	-1818	14552

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

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

Since none of the possible rational roots are actual roots, this means that the polynomial either has irrational roots or complex roots.

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