SOLUTION: 1. Use synthetic division to divide the polynomial 2x^3 + 7x^2 – 5 by 3+x and write the quotient and the remainder 2. Consider the polynomial f(x)= 4x^4+5x^3 + 7x^2 – 34x+8 B

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Question 149802: 1. Use synthetic division to divide the polynomial 2x^3 + 7x^2 – 5 by 3+x and write the quotient and the remainder
2. Consider the polynomial f(x)= 4x^4+5x^3 + 7x^2 – 34x+8
By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.


Find all of the zeros of the given polynomial. Show procedure

Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
# 1

Let's simplify this expression using synthetic division


Start with the given expression

First lets find our test zero:

Set the denominator equal to zero

Solve for x.

so our test zero is -3


Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.(note: remember if a polynomial goes from to there is a zero coefficient for . This is simply because really looks like
-3|270-5
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 2)
-3|270-5
|
2

Multiply -3 by 2 and place the product (which is -6) right underneath the second coefficient (which is 7)
-3|270-5
|-6
2

Add -6 and 7 to get 1. Place the sum right underneath -6.
-3|270-5
|-6
21

Multiply -3 by 1 and place the product (which is -3) right underneath the third coefficient (which is 0)
-3|270-5
|-6-3
21

Add -3 and 0 to get -3. Place the sum right underneath -3.
-3|270-5
|-6-3
21-3

Multiply -3 by -3 and place the product (which is 9) right underneath the fourth coefficient (which is -5)
-3|270-5
|-6-39
21-3

Add 9 and -5 to get 4. Place the sum right underneath 9.
-3|270-5
|-6-39
21-34


Since the last column adds to 4, we have a remainder of 4. This means is not a factor of

Now lets look at the bottom row of coefficients:


The first 3 coefficients (2,1,-3) form the quotient



and the last coefficient 4, is the remainder, which is placed over like this





Putting this altogether, we get:



So

which looks like this in remainder form:
remainder 4



-----------------------------------
Answer:

So the quotient is and the remainder is 4



# 2


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|457-348
| 27/221/4-115/8
4721/2-115/4-51/8

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


------------------------------------------------------


Let's make the synthetic division table for the function given the possible zero :
1/4|457-348
| 13/217/8-255/32
4617/2-255/81/32

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


------------------------------------------------------


Let's make the synthetic division table for the function given the possible zero :
2|457-348
| 8266664
413333272

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


------------------------------------------------------


Let's make the synthetic division table for the function given the possible zero :
4|457-348
| 16843641320
421913301328

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


------------------------------------------------------


Let's make the synthetic division table for the function given the possible zero :
8|457-348
| 32296242419120
437303239019128

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


------------------------------------------------------


Let's make the synthetic division table for the function given the possible zero :
-1|457-348
| -4-1-640
416-4048

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


------------------------------------------------------


Let's make the synthetic division table for the function given the possible zero :
-1/2|457-348
| -2-3/2-11/4147/8
4311/2-147/4211/8

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


------------------------------------------------------


Let's make the synthetic division table for the function given the possible zero :
-1/4|457-348
| -1-1-3/271/8
446-71/2135/8

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


------------------------------------------------------


Let's make the synthetic division table for the function given the possible zero :
-2|457-348
| -86-26120
4-313-60128

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


------------------------------------------------------


Let's make the synthetic division table for the function given the possible zero :
-4|457-348
| -1644-204952
4-1151-238960

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


------------------------------------------------------


Let's make the synthetic division table for the function given the possible zero :
-8|457-348
| -32216-178414544
4-27223-181814552

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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