SOLUTION: I need help with this question: use synthetic division to complete the indicated factorization x^3-37x+84=(x-4)( )

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Question 86302: I need help with this question: use synthetic division to complete the indicated factorization x^3-37x+84=(x-4)( )
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Let P be the unknown polynomial
Lets assume that is a factor

Start with the given equation

Divide both sides by


Now lets divide

First lets find our test zero:

Set the denominator equal to zero
Solve for x.

so our test zero is 4


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.(note: remember if a polynomial goes from to there is a zero coefficient for . This is simply because really looks like
4|10-3784
|

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
4|10-3784
|
1

Multiply 4 by 1 and place the product (which is 4) right underneath the second coefficient (which is 0)
4|10-3784
|4
1

Add 4 and 0 to get 4. Place the sum right underneath 4.
4|10-3784
|4
14

Multiply 4 by 4 and place the product (which is 16) right underneath the third coefficient (which is -37)
4|10-3784
|416
14

Add 16 and -37 to get -21. Place the sum right underneath 16.
4|10-3784
|416
14-21

Multiply 4 by -21 and place the product (which is -84) right underneath the fourth coefficient (which is 84)
4|10-3784
|416-84
14-21

Add -84 and 84 to get 0. Place the sum right underneath -84.
4|10-3784
|416-84
14-210

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 (1,4,-21) form the quotient



So

So the unknown polynomial P is



Now multiply both sides by



So the expression factors to this


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