SOLUTION: don't know how to check my amswers for these.... simplify: (3x^3-x^2-4x+1)/(x-1) factor: 18x+x^2-11x factor: 6x^2+x-2 factor: 12x^3+31x^2+20x

Question 149769: don't know how to check my amswers for these....
simplify: (3x^3-x^2-4x+1)/(x-1)
factor: 18x+x^2-11x
factor: 6x^2+x-2
factor: 12x^3+31x^2+20x
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
# 1

Let's simplify this expression using synthetic division

Start with the given expression

First lets find our test zero:

Set the denominator equal to zero

Solve for x.

so 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 numerator to the right of the test zero.

1	\|	3	-1	-4	1
	\|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 3)

1	\|	3	-1	-4	1
	\|
		3

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

1	\|	3	-1	-4	1
	\|		3
		3

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

1	\|	3	-1	-4	1
	\|		3
		3	2

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

1	\|	3	-1	-4	1
	\|		3	2
		3	2

Add 2 and -4 to get -2. Place the sum right underneath 2.

1	\|	3	-1	-4	1
	\|		3	2
		3	2	-2

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

1	\|	3	-1	-4	1
	\|		3	2	-2
		3	2	-2

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

1	\|	3	-1	-4	1
	\|		3	2	-2
		3	2	-2	-1

Since the last column adds to -1, we have a remainder of -1. This means is not a factor of
Now lets look at the bottom row of coefficients:

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

and the last coefficient -1, is the remainder, which is placed over like this

Putting this altogether, we get:

Start with the given expression.

Combine like terms.

Factor out the GCF

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

Answer:
So factors to

# 3

Looking at we can see that the first term is and the last term is where the coefficients are 6 and -2 respectively.

Now multiply the first coefficient 6 and the last coefficient -2 to get -12. Now what two numbers multiply to -12 and add to the middle coefficient 1? Let's list all of the factors of -12:

Factors of -12:
1,2,3,4,6,12

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

These factors pair up and multiply to -12
(1)*(-12)
(2)*(-6)
(3)*(-4)
(-1)*(12)
(-2)*(6)
(-3)*(4)

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	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

From this list we can see that -3 and 4 add up to 1 and multiply to -12

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 )

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

Answer:
So factors to

# 4

Start with the given expression

Factor out the GCF

Now let's focus on the inner expression

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

Looking at we can see that the first term is and the last term is where the coefficients are 12 and 20 respectively.

Now multiply the first coefficient 12 and the last coefficient 20 to get 240. Now what two numbers multiply to 240 and add to the middle coefficient 31? Let's list all of the factors of 240:

Factors of 240:
1,2,3,4,5,6,8,10,12,15,16,20,24,30,40,48,60,80,120,240

-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-16,-20,-24,-30,-40,-48,-60,-80,-120,-240 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 240
1*240
2*120
3*80
4*60
5*48
6*40
8*30
10*24
12*20
15*16
(-1)*(-240)
(-2)*(-120)
(-3)*(-80)
(-4)*(-60)
(-5)*(-48)
(-6)*(-40)
(-8)*(-30)
(-10)*(-24)
(-12)*(-20)
(-15)*(-16)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to 31? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 31

First Number	Second Number	Sum
1	240	1+240=241
2	120	2+120=122
3	80	3+80=83
4	60	4+60=64
5	48	5+48=53
6	40	6+40=46
8	30	8+30=38
10	24	10+24=34
12	20	12+20=32
15	16	15+16=31
-1	-240	-1+(-240)=-241
-2	-120	-2+(-120)=-122
-3	-80	-3+(-80)=-83
-4	-60	-4+(-60)=-64
-5	-48	-5+(-48)=-53
-6	-40	-6+(-40)=-46
-8	-30	-8+(-30)=-38
-10	-24	-10+(-24)=-34
-12	-20	-12+(-20)=-32
-15	-16	-15+(-16)=-31

From this list we can see that 15 and 16 add up to 31 and multiply to 240

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

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