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

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Question 140329: Find all the zeros of

Answer by jim_thompson5910(35256) (Show Source):
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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 -5 3 -3 1 2
|

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

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

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

2 | 2 -5 3 -3 1 2
| 4
2

Add 4 and -5 to get -1. Place the sum right underneath 4.

2 | 2 -5 3 -3 1 2
| 4
2 -1

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

2 | 2 -5 3 -3 1 2
| 4 -2
2 -1

Add -2 and 3 to get 1. Place the sum right underneath -2.

2 | 2 -5 3 -3 1 2
| 4 -2
2 -1 1

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

2 | 2 -5 3 -3 1 2
| 4 -2 2
2 -1 1

Add 2 and -3 to get -1. Place the sum right underneath 2.

2 | 2 -5 3 -3 1 2
| 4 -2 2
2 -1 1 -1

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

2 | 2 -5 3 -3 1 2
| 4 -2 2 -2
2 -1 1 -1

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

2 | 2 -5 3 -3 1 2
| 4 -2 2 -2
2 -1 1 -1 -1

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

2 | 2 -5 3 -3 1 2
| 4 -2 2 -2 -2
2 -1 1 -1 -1

Add -2 and 2 to get 0. Place the sum right underneath -2.

2 | 2 -5 3 -3 1 2
| 4 -2 2 -2 -2
2 -1 1 -1 -1 0

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 -1 1 -1 -1
|

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

1 | 2 -1 1 -1 -1
|
2

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

1 | 2 -1 1 -1 -1
| 2
2

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

1 | 2 -1 1 -1 -1
| 2
2 1

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

1 | 2 -1 1 -1 -1
| 2 1
2 1

Add 1 and 1 to get 2. Place the sum right underneath 1.

1 | 2 -1 1 -1 -1
| 2 1
2 1 2

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

1 | 2 -1 1 -1 -1
| 2 1 2
2 1 2

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

1 | 2 -1 1 -1 -1
| 2 1 2
2 1 2 1

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

1 | 2 -1 1 -1 -1
| 2 1 2 1
2 1 2 1

Add 1 and -1 to get 0. Place the sum right underneath 1.

1 | 2 -1 1 -1 -1
| 2 1 2 1
2 1 2 1 0

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 | 2 1 2 1
|

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

-1/2 | 2 1 2 1
|
2

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

-1/2 | 2 1 2 1
| -1
2

Add -1 and 1 to get 0. Place the sum right underneath -1.

-1/2 | 2 1 2 1
| -1
2 0

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

-1/2 | 2 1 2 1
| -1 0
2 0

Add 0 and 2 to get 2. Place the sum right underneath 0.

-1/2 | 2 1 2 1
| -1 0
2 0 2

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

-1/2 | 2 1 2 1
| -1 0 -1
2 0 2

Add -1 and 1 to get 0. Place the sum right underneath -1.

-1/2 | 2 1 2 1
| -1 0 -1
2 0 2 0

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

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Answer:

So the zeros of are

, , , , or