SOLUTION: Find all the zeros of {{{f(x)=2x^5-5x^4+3x^3-3x^2+x+2}}}

Algebra.Com
Question 140329: Find all the zeros of


Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
First graph the function





From the graph, we can see that there is a zero at . So our test zero is 2



Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the function to the right of the test zero.
2|2-53-312
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 2)
2|2-53-312
|
2

Multiply 2 by 2 and place the product (which is 4) right underneath the second coefficient (which is -5)
2|2-53-312
|4
2

Add 4 and -5 to get -1. Place the sum right underneath 4.
2|2-53-312
|4
2-1

Multiply 2 by -1 and place the product (which is -2) right underneath the third coefficient (which is 3)
2|2-53-312
|4-2
2-1

Add -2 and 3 to get 1. Place the sum right underneath -2.
2|2-53-312
|4-2
2-11

Multiply 2 by 1 and place the product (which is 2) right underneath the fourth coefficient (which is -3)
2|2-53-312
|4-22
2-11

Add 2 and -3 to get -1. Place the sum right underneath 2.
2|2-53-312
|4-22
2-11-1

Multiply 2 by -1 and place the product (which is -2) right underneath the fifth coefficient (which is 1)
2|2-53-312
|4-22-2
2-11-1

Add -2 and 1 to get -1. Place the sum right underneath -2.
2|2-53-312
|4-22-2
2-11-1-1

Multiply 2 by -1 and place the product (which is -2) right underneath the sixth coefficient (which is 2)
2|2-53-312
|4-22-2-2
2-11-1-1

Add -2 and 2 to get 0. Place the sum right underneath -2.
2|2-53-312
|4-22-2-2
2-11-1-10

Since the last column adds to zero, we have a remainder of zero. This means is a factor of

Now lets look at the bottom row of coefficients:

The first 5 coefficients (2,-1,1,-1,-1) form the quotient




So

You can use this online polynomial division calculator to check your work

Basically factors to

Now lets break down further




Now let's graph the function





From the graph, we can see that there is a zero at . So our test zero is 1. So this time our test zero is 1



Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the function to the right of the test zero.
1|2-11-1-1
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 2)
1|2-11-1-1
|
2

Multiply 1 by 2 and place the product (which is 2) right underneath the second coefficient (which is -1)
1|2-11-1-1
|2
2

Add 2 and -1 to get 1. Place the sum right underneath 2.
1|2-11-1-1
|2
21

Multiply 1 by 1 and place the product (which is 1) right underneath the third coefficient (which is 1)
1|2-11-1-1
|21
21

Add 1 and 1 to get 2. Place the sum right underneath 1.
1|2-11-1-1
|21
212

Multiply 1 by 2 and place the product (which is 2) right underneath the fourth coefficient (which is -1)
1|2-11-1-1
|212
212

Add 2 and -1 to get 1. Place the sum right underneath 2.
1|2-11-1-1
|212
2121

Multiply 1 by 1 and place the product (which is 1) right underneath the fifth coefficient (which is -1)
1|2-11-1-1
|2121
2121

Add 1 and -1 to get 0. Place the sum right underneath 1.
1|2-11-1-1
|2121
21210

Since the last column adds to zero, we have a remainder of zero. This means is a factor of

Now lets look at the bottom row of coefficients:

The first 4 coefficients (2,1,2,1) form the quotient




So

You can use this online polynomial division calculator to check your work

Basically factors to

Now lets break down further





Now let's graph the function





From the graph, we can see that there is a zero at . So our test zero is . So this time our test zero is 1





Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the function to the right of the test zero.
-1/2|2121
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 2)
-1/2|2121
|
2

Multiply -1/2 by 2 and place the product (which is -1) right underneath the second coefficient (which is 1)
-1/2|2121
|-1
2

Add -1 and 1 to get 0. Place the sum right underneath -1.
-1/2|2121
|-1
20

Multiply -1/2 by 0 and place the product (which is 0) right underneath the third coefficient (which is 2)
-1/2|2121
|-10
20

Add 0 and 2 to get 2. Place the sum right underneath 0.
-1/2|2121
|-10
202

Multiply -1/2 by 2 and place the product (which is -1) right underneath the fourth coefficient (which is 1)
-1/2|2121
|-10-1
202

Add -1 and 1 to get 0. Place the sum right underneath -1.
-1/2|2121
|-10-1
2020

Since the last column adds to zero, we have a remainder of zero. This means is a factor of

Now lets look at the bottom row of coefficients:

The first 3 coefficients (2,0,2) form the quotient




Notice in the denominator , the x term has a coefficient of 2, so we need to divide the quotient by 2 like this:


So

You can use this online polynomial division calculator to check your work

Basically factors to

Now lets break down further



Set the factor equal to zero


Subtract 1 from both sides


Take the square root of both sides


Simplify

or




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

Answer:


So the zeros of are


, , , , or
RELATED QUESTIONS

list all the possible rational zeros of... (answered by Edwin McCravy)
Please help me find all real zeros for this function f(x)= 2x^4+3x^3-3x^2+3x-5 (answered by lynnlo,stanbon)
Find all the rational zeros of the function... (answered by stanbon,scott8148)
find all the REAL zeros of f(x) =... (answered by stanbon)
f(x)=3x^3+11x^2+5x-3 find all the zeros in this polynomial... (answered by stanbon)
find all the real zeros and state their multiplicities.... (answered by nyc_function)
Find all zeros of... (answered by MathLover1)
find all solution ( real and complex) to the equation: x^4+3x^2=x^3 find all zeros... (answered by Fombitz)
list all the possible rational zeros of... (answered by solver91311)