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

Algebra ->  Polynomials-and-rational-expressions -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
# 1

Let's simplify this expression using synthetic division


Start with the given expression %282x%5E3+%2B+7x%5E2+-+5%29%2F%283%2Bx%29

First lets find our test zero:

3%2Bx=0 Set the denominator 3%2Bx equal to zero

x=-3 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 7x%5E2 to -5x%5E0 there is a zero coefficient for x%5E1. This is simply because 2x%5E3+%2B+7x%5E2+-+5 really looks like 2x%5E3%2B7x%5E2%2B0x%5E1%2B-5x%5E0
-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 3%2Bx is not a factor of 2x%5E3+%2B+7x%5E2+-+5

Now lets look at the bottom row of coefficients:


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

2x%5E2+%2B+x+-+3

and the last coefficient 4, is the remainder, which is placed over 3%2Bx like this

4%2F%283%2Bx%29



Putting this altogether, we get:

2x%5E2+%2B+x+-+3%2B4%2F%283%2Bx%29

So %282x%5E3+%2B+7x%5E2+-+5%29%2F%283%2Bx%29=2x%5E2+%2B+x+-+3%2B4%2F%283%2Bx%29

which looks like this in remainder form:
%282x%5E3+%2B+7x%5E2+-+5%29%2F%283%2Bx%29=2x%5E2+%2B+x+-+3 remainder 4



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

So the quotient is 2x%5E2+%2B+x+-+3 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 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero 1%2F2:
1/2|457-348
| 27/221/4-115/8
4721/2-115/4-51/8

Since the remainder -51%2F8 (the right most entry in the last row) is not equal to zero, this means that 1%2F2 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8


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Let's make the synthetic division table for the function 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero 1%2F4:
1/4|457-348
| 13/217/8-255/32
4617/2-255/81/32

Since the remainder 1%2F32 (the right most entry in the last row) is not equal to zero, this means that 1%2F4 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8


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Let's make the synthetic division table for the function 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero 2:
2|457-348
| 8266664
413333272

Since the remainder 72 (the right most entry in the last row) is not equal to zero, this means that 2 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8


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Let's make the synthetic division table for the function 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero 4:
4|457-348
| 16843641320
421913301328

Since the remainder 1328 (the right most entry in the last row) is not equal to zero, this means that 4 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8


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Let's make the synthetic division table for the function 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero 8:
8|457-348
| 32296242419120
437303239019128

Since the remainder 19128 (the right most entry in the last row) is not equal to zero, this means that 8 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8


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Let's make the synthetic division table for the function 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero -1:
-1|457-348
| -4-1-640
416-4048

Since the remainder 48 (the right most entry in the last row) is not equal to zero, this means that -1 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8


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Let's make the synthetic division table for the function 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero -1%2F2:
-1/2|457-348
| -2-3/2-11/4147/8
4311/2-147/4211/8

Since the remainder 211%2F8 (the right most entry in the last row) is not equal to zero, this means that -1%2F2 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8


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Let's make the synthetic division table for the function 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero -1%2F4:
-1/4|457-348
| -1-1-3/271/8
446-71/2135/8

Since the remainder 135%2F8 (the right most entry in the last row) is not equal to zero, this means that -1%2F4 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8


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


Let's make the synthetic division table for the function 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero -2:
-2|457-348
| -86-26120
4-313-60128

Since the remainder 128 (the right most entry in the last row) is not equal to zero, this means that -2 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8


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Let's make the synthetic division table for the function 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero -4:
-4|457-348
| -1644-204952
4-1151-238960

Since the remainder 960 (the right most entry in the last row) is not equal to zero, this means that -4 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8


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Let's make the synthetic division table for the function 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8 given the possible zero -8:
-8|457-348
| -32216-178414544
4-27223-181814552

Since the remainder 14552 (the right most entry in the last row) is not equal to zero, this means that -8 is not a zero of 4x%5E4%2B5x%5E3%2B7x%5E2-34x%2B8



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