SOLUTION: Find the factorization of {{{8x^3+34x^2-39x-45}}}

Question 138660: Find the factorization of
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
First graph the function

Graph of

From the graph we can see that the function has a zero at . So our test zero is -5. So our first factor is

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.

-5	\|	8	34	-39	-45
	\|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 8)

-5	\|	8	34	-39	-45
	\|
		8

Multiply -5 by 8 and place the product (which is -40) right underneath the second coefficient (which is 34)

-5	\|	8	34	-39	-45
	\|		-40
		8

Add -40 and 34 to get -6. Place the sum right underneath -40.

-5	\|	8	34	-39	-45
	\|		-40
		8	-6

Multiply -5 by -6 and place the product (which is 30) right underneath the third coefficient (which is -39)

-5	\|	8	34	-39	-45
	\|		-40	30
		8	-6

Add 30 and -39 to get -9. Place the sum right underneath 30.

-5	\|	8	34	-39	-45
	\|		-40	30
		8	-6	-9

Multiply -5 by -9 and place the product (which is 45) right underneath the fourth coefficient (which is -45)

-5	\|	8	34	-39	-45
	\|		-40	30	45
		8	-6	-9

Add 45 and -45 to get 0. Place the sum right underneath 45.

-5	\|	8	34	-39	-45
	\|		-40	30	45
		8	-6	-9	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 (8,-6,-9) form the quotient

So

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

Basically factors to

Now lets break down further

Looking at we can see that the first term is and the last term is where the coefficients are 8 and -9 respectively.

Now multiply the first coefficient 8 and the last coefficient -9 to get -72. Now what two numbers multiply to -72 and add to the middle coefficient -6? Let's list all of the factors of -72:

Factors of -72:
1,2,3,4,6,8,9,12,18,24,36,72

-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -72
(1)*(-72)
(2)*(-36)
(3)*(-24)
(4)*(-18)
(6)*(-12)
(8)*(-9)
(-1)*(72)
(-2)*(36)
(-3)*(24)
(-4)*(18)
(-6)*(12)
(-8)*(9)

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

Now which of these pairs add to -6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -6

First Number	Second Number	Sum
1	-72	1+(-72)=-71
2	-36	2+(-36)=-34
3	-24	3+(-24)=-21
4	-18	4+(-18)=-14
6	-12	6+(-12)=-6
8	-9	8+(-9)=-1
-1	72	-1+72=71
-2	36	-2+36=34
-3	24	-3+24=21
-4	18	-4+18=14
-6	12	-6+12=6
-8	9	-8+9=1

From this list we can see that 6 and -12 add up to -6 and multiply to -72

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

So this also means that factors to (since is equivalent to )

So factors to

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

So the original polynomial factors to

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