Is the given a factor of the polynomial?

Question

Directions say Use synthetic division to determine whether x-c is a factor of the given polynomial.

problem: p(x)=x³-2x²-3x+6; x-√3

problem: p(x)=x⁴-x³-5x²-x-6; x-i

Thank you!

Answer 1

For x – √3 to be a factor, you must have x = √3 as a zero. Using this information, do the synthetic division with  x = √3 as the test zero on the left: Start out with the synthetic division algorithm.

Start as usual by bringing down the 1:

√3 |     1        -2           -3        6

     |  ___________________                                              

            1    

Multiply the 1 by √3 and put it diagonally above the 1 under the -2. 
Add -2 and √3, getting -2+√3 and write this on the bottom of the line.

 

√3 |     1        -2           -3        6

     |                 √3                              

            1     -2+√3     

 

Multiply the -2+√3 by √3 and put it diagonally above the -2+√3 under the -3.
Add -3 and -2√3+3, getting -2√3 and write this on the bottom of the line.

 

√3 |     1        -2           -3        6

     |                 √3         -2√3+3          

            1     -2+√3     -2√3    

 

Multiply the -2√3 by √3 and put it diagonally above the -2√3 under the 6. 
Add 6 and -6, getting 0 as ramainder and write this on the bottom of the line.

 

√3 |     1        -2                -3             6

     |                 √3         -2√3+3        -6

            1      -2+√3     -2√3              0

Since the remainder is zero, then  x = √3 is indeed a zero of x³ - 2x²  – 3x + 6,

so: x - √3 is a factor of x³ - 2x²  – 3x + 6.

 

Answer 2

For x – i to be a factor, you must have x = i as a zero. Using this information, do the synthetic division with  x = i as the test zero on the left:

 

i    |     1        -1           -5             -1      -6

     |                  i           -i-1         1-6i      6

             1     -1+i         -i-6        -6i        0

Since the remainder is zero, then  x = i is indeed a zero of x⁴ - x³  – 5x² -x - 6,

so: x – i is a factor of  x⁴ - x³  – 5x² -x - 6.