Dividing Polynomials

Question

(35w³⁺18w²⁺32w+35)/(5w+4)

Answer 1

(35w³ + 18w² + 32w + 35) / (5w + 4)

 

 

We will use a method known as synthetic division.  We divide the coefficients of the polynomial by the root of it factor.  The digit in the quotient is the remainder.

 

The coefficient of the polynomial is  35  18  32   35.

The factor of the polynomial is (5w + 4).  So the root is 4/5.

 

 

Now, we set up the division.

 

-4/5  |   35   18    32    35

 

 

_______________________

 

 

 

 

Step one:  Bring down the first coefficient under the line.

 

-4/5  |   35   18    32    35
 
 
_______________________

            35

 

 

Step two:  Multiply the root by the first digit under the line.  Place the product as the addend in the next column.

 

-4/5  |   35     18     32     35
 
                    -28     
__________________________

            35   

 

 

Step three: Add the two numbers in the second column.  

 

 

-4/5  |   35     18     32     35
 
                    -28
__________________________
            35    -10

 

 

Repeat the above steps for the succeeding columns.

 

 

-4/5  |   35       18            32             35
 
                      -28              8            -32
____________________________________
            35      -10             40             3

 

 

The quotient is   35w² - 10w + 40

 

Divide the quotient by 5 because we need the first term in this quotient to be 7w².

 

 

Quotient = 7w² - 2w + 8

The remainder is  3

 

 

Answer 2

There are a couple of ways to do this...one is long division...

 

5w+4 √ 35w^3+18w^2+32w+35...

 

we need 7w^2, so we subtract off 35w^3+28w^2, leaving -10w^2+32w+35...

 

Next we need -2w, and subtract off -10w^2-8w, leaving 40w+35...

 

Lastly we need 8 more, and subtract off the last 40w+32...leaving a remainder of 3, that can't be divided further by 5w+4...

 

Thus (5w+4)*(7w^2-2w+8)+3 = 35w^3+18w^2+32w+35.

 

 

There is a nice trick for doing this neatly...

 

             7

5w+4 | 35 18 32 35

            35 28

                -10 32 35

 

             7  -2
5w+4 | 35 18 32 35
           35 28
               -10 32 35

               -10  -8

                     40 35

 

              7 -2  8
5w+4 | 35 18 32 35
           35 28
               -10 32 35
               -10 -8
                     40 35

                     40 32

                           3remaining

 

Hope that helps!