SOLUTION: Factor the polynomial below completely, given the bionomial folowing it is a factor of thepoynomial x^3-4x^2+x+6, x+1

Algebra ->  Graphs -> SOLUTION: Factor the polynomial below completely, given the bionomial folowing it is a factor of thepoynomial x^3-4x^2+x+6, x+1       Log On


   

Question 149491: Factor the polynomial below completely, given the bionomial folowing it is a factor of thepoynomial x^3-4x^2+x+6, x+1

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's simplify this expression using synthetic division


Start with the given expression %28x%5E3+-+4x%5E2+%2B+x+%2B+6%29%2F%28x%2B1%29

First lets find our test zero:

x%2B1=0 Set the denominator x%2B1 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|1-416
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
-1|1-416
|
1

Multiply -1 by 1 and place the product (which is -1) right underneath the second coefficient (which is -4)
-1|1-416
|-1
1

Add -1 and -4 to get -5. Place the sum right underneath -1.
-1|1-416
|-1
1-5

Multiply -1 by -5 and place the product (which is 5) right underneath the third coefficient (which is 1)
-1|1-416
|-15
1-5

Add 5 and 1 to get 6. Place the sum right underneath 5.
-1|1-416
|-15
1-56

Multiply -1 by 6 and place the product (which is -6) right underneath the fourth coefficient (which is 6)
-1|1-416
|-15-6
1-56

Add -6 and 6 to get 0. Place the sum right underneath -6.
-1|1-416
|-15-6
1-560


Now lets look at the bottom row of coefficients:

The first 3 coefficients (1,-5,6) form the quotient

x%5E2+-+5x+%2B+6


So %28x%5E3+-+4x%5E2+%2B+x+%2B+6%29%2F%28x%2B1%29=x%5E2+-+5x+%2B+6

Basically x%5E3+-+4x%5E2+%2B+x+%2B+6 factors to %28x%2B1%29%28x%5E2+-+5x+%2B+6%29

Now lets break x%5E2+-+5x+%2B+6 down further




Looking at the expression x%5E2-5x%2B6, we can see that the first coefficient is 1, the second coefficient is -5, and the last term is 6.


Now multiply the first coefficient 1 by the last term 6 to get %281%29%286%29=6.


Now the question is: what two whole numbers multiply to 6 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of 6 (the previous product).


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


Note: list the negative of each factor. This will allow us to find all possible combinations.


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:


First NumberSecond NumberSum
161+6=7
232+3=5
-1-6-1+(-6)=-7
-2-3-2+(-3)=-5



From the table, we can see that the two numbers -2 and -3 add to -5 (the middle coefficient).


So the two numbers -2 and -3 both multiply to 6 and add to -5


Now replace the middle term -5x with -2x-3x. Remember, -2 and -3 add to -5. So this shows us that -2x-3x=-5x.


x%5E2%2Bhighlight%28-2x-3x%29%2B6 Replace the second term -5x with -2x-3x.


%28x%5E2-2x%29%2B%28-3x%2B6%29 Group the terms into two pairs.


x%28x-2%29%2B%28-3x%2B6%29 Factor out the GCF x from the first group.


x%28x-2%29-3%28x-2%29 Factor out 3 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28x-3%29%28x-2%29 Combine like terms. Or factor out the common term x-2



So x%5E2-5x%2B6 factors to %28x-3%29%28x-2%29.


So %28x%2B1%29%28x%5E2+-+5x+%2B+6%29 now becomes %28x%2B1%29%28x-3%29%28x-2%29

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Answer:


So x%5E3-4x%5E2%2Bx%2B6 completely factors to %28x%2B1%29%28x-3%29%28x-2%29