SOLUTION: Can I please get assistance with the following problem? factor f(x) into linear factors given that k is a zero of f(x) f(x)=x^3-6x^2-25x+150;k=5 THANK YOU!!

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Question 140955This question is from textbook college algebra
: Can I please get assistance with the following problem?
factor f(x) into linear factors given that k is a zero of f(x)
f(x)=x^3-6x^2-25x+150;k=5
THANK YOU!! This question is from textbook college algebra

Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
Since k=5 is a zero, this means that 5 is a test zero

So break down by using synthetic division

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.

5 | 1 -6 -25 150
|

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

5 | 1 -6 -25 150
|
1

Multiply 5 by 1 and place the product (which is 5) right underneath the second coefficient (which is -6)

5 | 1 -6 -25 150
| 5
1

Add 5 and -6 to get -1. Place the sum right underneath 5.

5 | 1 -6 -25 150
| 5
1 -1

Multiply 5 by -1 and place the product (which is -5) right underneath the third coefficient (which is -25)

5 | 1 -6 -25 150
| 5 -5
1 -1

Add -5 and -25 to get -30. Place the sum right underneath -5.

5 | 1 -6 -25 150
| 5 -5
1 -1 -30

Multiply 5 by -30 and place the product (which is -150) right underneath the fourth coefficient (which is 150)

5 | 1 -6 -25 150
| 5 -5 -150
1 -1 -30

Add -150 and 150 to get 0. Place the sum right underneath -150.

5 | 1 -6 -25 150
| 5 -5 -150
1 -1 -30 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 (1,-1,-30) form the quotient

So

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

Basically factors to

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Now lets break down further

Looking at we can see that the first term is and the last term is where the coefficients are 1 and -30 respectively.

Now multiply the first coefficient 1 and the last coefficient -30 to get -30. Now what two numbers multiply to -30 and add to the middle coefficient -1? Let's list all of the factors of -30:

Factors of -30:
1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -30
(1)*(-30)
(2)*(-15)
(3)*(-10)
(5)*(-6)
(-1)*(30)
(-2)*(15)
(-3)*(10)
(-5)*(6)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1

First Number Second Number Sum
1 -30 1+(-30)=-29
2 -15 2+(-15)=-13
3 -10 3+(-10)=-7
5 -6 5+(-6)=-1
-1 30 -1+30=29
-2 15 -2+15=13
-3 10 -3+10=7
-5 6 -5+6=1

From this list we can see that 5 and -6 add up to -1 and multiply to -30

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to which means that factors to

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Answer:

So factors to