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

Algebra ->  Polynomials-and-rational-expressions -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
# 1


Let's simplify this expression using synthetic division


Start with the given expression %283x%5E3+-+x%5E2+-+4x+%2B+1%29%2F%28x-1%29

First lets find our test zero:

x-1=0 Set the denominator x-1 equal to zero

x=1 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|3-1-41
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 3)
1|3-1-41
|
3

Multiply 1 by 3 and place the product (which is 3) right underneath the second coefficient (which is -1)
1|3-1-41
|3
3

Add 3 and -1 to get 2. Place the sum right underneath 3.
1|3-1-41
|3
32

Multiply 1 by 2 and place the product (which is 2) right underneath the third coefficient (which is -4)
1|3-1-41
|32
32

Add 2 and -4 to get -2. Place the sum right underneath 2.
1|3-1-41
|32
32-2

Multiply 1 by -2 and place the product (which is -2) right underneath the fourth coefficient (which is 1)
1|3-1-41
|32-2
32-2

Add -2 and 1 to get -1. Place the sum right underneath -2.
1|3-1-41
|32-2
32-2-1


Since the last column adds to -1, we have a remainder of -1. This means x-1 is not a factor of 3x%5E3+-+x%5E2+-+4x+%2B+1
Now lets look at the bottom row of coefficients:

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

3x%5E2+%2B+2x+-+2

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

-1%2F%28x-1%29



Putting this altogether, we get:

3x%5E2+%2B+2x+-+2%2B-1%2F%28x-1%29

So %283x%5E3+-+x%5E2+-+4x+%2B+1%29%2F%28x-1%29=3x%5E2+%2B+2x+-+2%2B-1%2F%28x-1%29






18x%2Bx%5E2-11x Start with the given expression.


x%5E2%2B7x Combine like terms.


x%28x%2B7%29 Factor out the GCF x



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Answer:
So 18x%2Bx%5E2-11x factors to x%28x%2B7%29






# 3



Looking at 6x%5E2%2Bx-2 we can see that the first term is 6x%5E2 and the last term is -2 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 NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+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 6x%5E2%2Bx-2, replace x with -3x%2B4x (notice -3x%2B4x adds up to x. So it is equivalent to x)

6x%5E2%2Bhighlight%28-3x%2B4x%29%2B-2


Now let's factor 6x%5E2-3x%2B4x-2 by grouping:


%286x%5E2-3x%29%2B%284x-2%29 Group like terms


3x%282x-1%29%2B2%282x-1%29 Factor out the GCF of 3x out of the first group. Factor out the GCF of 2 out of the second group


%283x%2B2%29%282x-1%29 Since we have a common term of 2x-1, we can combine like terms

So 6x%5E2-3x%2B4x-2 factors to %283x%2B2%29%282x-1%29


So this also means that 6x%5E2%2Bx-2 factors to %283x%2B2%29%282x-1%29 (since 6x%5E2%2Bx-2 is equivalent to 6x%5E2-3x%2B4x-2)



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Answer:
So 6x%5E2%2Bx-2 factors to %283x%2B2%29%282x-1%29




# 4



12x%5E3%2B31x%5E2%2B20x Start with the given expression


x%2812x%5E2%2B31x%2B20%29 Factor out the GCF x


Now let's focus on the inner expression 12x%5E2%2B31x%2B20




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Looking at 12x%5E2%2B31x%2B20 we can see that the first term is 12x%5E2 and the last term is 20 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 NumberSecond NumberSum
12401+240=241
21202+120=122
3803+80=83
4604+60=64
5485+48=53
6406+40=46
8308+30=38
102410+24=34
122012+20=32
151615+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 12x%5E2%2B31x%2B20, replace 31x with 15x%2B16x (notice 15x%2B16x adds up to 31x. So it is equivalent to 31x)

12x%5E2%2Bhighlight%2815x%2B16x%29%2B20


Now let's factor 12x%5E2%2B15x%2B16x%2B20 by grouping:


%2812x%5E2%2B15x%29%2B%2816x%2B20%29 Group like terms


3x%284x%2B5%29%2B4%284x%2B5%29 Factor out the GCF of 3x out of the first group. Factor out the GCF of 4 out of the second group


%283x%2B4%29%284x%2B5%29 Since we have a common term of 4x%2B5, we can combine like terms

So 12x%5E2%2B15x%2B16x%2B20 factors to %283x%2B4%29%284x%2B5%29


So this also means that 12x%5E2%2B31x%2B20 factors to %283x%2B4%29%284x%2B5%29 (since 12x%5E2%2B31x%2B20 is equivalent to 12x%5E2%2B15x%2B16x%2B20)



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So our expression goes from x%2812x%5E2%2B31x%2B20%29 and factors further to x%283x%2B4%29%284x%2B5%29


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Answer:

So 12x%5E3%2B31x%5E2%2B20x factors to x%283x%2B4%29%284x%2B5%29