SOLUTION: Use the factor theorem and synthetic division to determine whether or not the first polynomial is a factor of the second. (x+1); x^4 - x^3 - 2x^2 + 7x + 7 I also worked on t

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Use the factor theorem and synthetic division to determine whether or not the first polynomial is a factor of the second. (x+1); x^4 - x^3 - 2x^2 + 7x + 7 I also worked on t      Log On


   

Question 920397: Use the factor theorem and synthetic division to determine whether or not the first polynomial is a factor of the second.
(x+1); x^4 - x^3 - 2x^2 + 7x + 7
I also worked on this myself and got this:
(-1)^4 - (-1)^3 - 2(-1)^2 + 7(-1) + 7, which turns into...
... 1 + 1 - 2 - 7 + 7 = 0, meaning (x + 1) IS a factor, I think? But I haven't gone into synthetic division, and I could be wrong. Please correct me if I am, thank you!
Found 2 solutions by josgarithmetic, Edwin McCravy:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Synthetic division gives the resulting quotient coefficients of 1,-2,0,7, and the remainder is 0.

x%2B1 is a factor of the polynomial.

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Here's what you are supposed to do:

Start with the coefficients of the terms
and this array:

-1| 1 -1 -2  7  7
  |______________
    


Bring down the 1 down below the line

-1| 1 -1 -2  7  7
  |                 
    1 

Multiply that 1 by the -1 at the far left, getting -1
Place it above and to the right of that 1:

-1| 1 -1 -2  7  7
  |   -1          
    1  

Add the -1 and the -1 getting -2 and write it below the line.

-1| 1 -1 -2  7  7
  |   -1         
    1 -2  

Multiply that -2 by the -1 at the far left, getting 2
Place it above and to the right of that -2:

-1| 1 -1 -2  7  7
  |   -1  2        
    1 -2  

Add the -2 and the 2 getting 0 and write it below the line.

-1| 1 -1 -2  7  7
  |   -1  2        
    1 -2  0  

Multiply that 0 by the -1 at the far left, getting 0
Place it above and to the right of that 0:

-1| 1 -1 -2  7  7
  |   -1  2  0    
    1 -2  0  

Add the 7 and the 0 getting 7 and write it below the line.

-1| 1 -1 -2  7  7
  |   -1  2  0    
    1 -2  0  7  

Multiply that 7 by the -1 at the far left, getting -7
Place it above and to the right of that 7:

-1| 1 -1 -2  7  7
  |   -1  2  0 -7
    1 -2  0  7  

Add the 7 and the -7 getting 0 and write it below the line.

-1| 1 -1 -2  7  7
  |   -1  2  0 -7
    1 -2  0  7  0

Now you've finished the synthetic division.  The 0 on the bottom
right tells you that the remainder is 0, and that x+1 is indeed a
factor.  The other numbers before that 0 on the far right,

    1 -2  0  7  

tell you the coefficients of the quotient you would get if you did 
the problem by long division instead of synthetic division.

The quotient is a polynomial of 1 degree less than the degree of
the original polynomial with those coefficients. The original
polynomial was of degree 4, so the quotient is of degree 3, so
using those numbers as coefficients, you have the quotient as:

    1x3-2x2+0x+7

or just

    x3-2x2+7

Which means you have now factored the polynomial as

(x+1)(x3-2x2+7)

Edwin