Hi guys! This is Claudine and I'm going to write about Synthetic Division, Remainder Theorem, and Factor Theorem.

S   Y   N   T   H   E   T   I   C         D   I   V   I   S   I   O   N
     
        It is a simplified method of dividing a polynomial P(x) by x-a, where a is any assigned number. By this method, we determine values of the coefficients of the quotient and the value of the remainder can be easily determined.

Example 1 : Divide  5x + x⁴  - 14x²  by x + 2 using synthetic division.

1st: Write the terms of the dividend in descending powers of the variable. 
2nd: Fill in missing terms using zero for the coefficients, then write the divisor in the form x - a


(x⁴ + 0x³ -14x² + 5x + 0) / (x-(-2))
Write the constant term a from the divisor on __| & write the coefficients from the dividend to the right of the symbol

-2|   1   0   -14   5   0

Bring down the 1st term in the divisor to the third row.

-2|  1   0   -14   5   0  
________________
      1

Multiply 1 (term in 3rd row) by -2 (the divisor). Then, write the product in the 2nd row under the 2nd term in the 1st row. Add the numbers in the column formed, and write the sum as the 2nd term in the Quotient row (3rd row)

-2|   1   0   -14   5   0
           -2                  
       1  -2

Multiply the last term on the right in the Quotient row (3rd row) by the divisor, write it under the next term in the top row, add, and write the sum in the quotient row. Continue this process until all of the terms in the top row have a number under them.

-2|    1   0    -14      5      0
            -2      4     20   -50
        1  -2   -10     25   -50

• The 3rd row is the quotient row with the last term being the remainder. If the last term is 0, then it has no remainder.
• The degree of the quotient polynomial is degree of dividend - 1 because we are dividing by a linear factor.
• The terms of the quotient row are the coefficients of the terms in the quotient polynomial.

The degree of the quotient polynomial here is 3. The quotient with remainder for 5x + x⁴  - 14x²  by x + 2 is:


(1)x³ - 2x² - 10x + 25 + R-50   or   x³ - 2x² - 10x + 25 - 50/x+2

Example 2 : This one has images to make it easier to understand. Divide 2x⁴ – 3x³ – 5x² + 3x + 8 by x – 2

 Write in this form:

First, carry down the "2" that indicates the leading coefficient:

Multiply the number on the left, and carry the result into the next column:

Add down the column:

Multiply by the number on the left, and carry the result into the next column:

Add down the column:

Multiply by the number on the left, and carry the result into the next column:

Add down the column:

Multiply by the number on the left, and carry the result into the next column:

Add down the column for the remainder:

This is the result:




Example 3 :  If you don't want to read or prefer learning by watching, here is a video on how to solve using synthetic division :P

R  E  M  A  I  N  D  E  R         T  H  E  O  R  E  M

If a is any constant and if a polynomial P(x) is divided by (x-a), the remainder is P(a).

Example : If P(x) = 2x³ - 3x² - x + 8 is divided by x + 1, then a = -1 
and the remainder = P(-1) = -2 - 3 + 1 + 8 = 4. That is.

P(x) / (x+1) = Q(x) + 4 / (x+1) , where Q(x) is a polynomial in x.

F   A   C   T   O   R          T   H   E   O   R   E   M

If a is a root of the equation P(x) = 0, if P(a) = 0, then (x-a) is a factor of P(x). Conversely, if (x-a) is a factor of P(x), then a is a root of P(x) =0, or P(a) = 0.

Example : Thus there are 3 roots of the equation P(x) = x³ + 4x² + x - 6 = 0, since 
P(1) = P(-2) = P(-3) = 0. Then (x - 1), (x + 2) and (x + 3) are factors of P(x).

APPLICATIONS:

1. Determine the remainder R in 2x³ + 3x² - 18x - 4 / (x - 2)

Answer: R = P(2) = (2)³ + (3)(2)² - (18)(2) - 4 = -12

2. Show that ( x - 3 ) is a factor of the polynomial P(x) = x⁴ - 4x³ - 7x² + 22x + 24

Answer: P(3) = 81 - 108 - 63 + 66 + 24 = 0. Therefore, (x - 3) is a factor of P(x), 3 is a zero of the Polynomial P(x), and 3 is a root of the equation P(x) = 0

3. Find values of p for which 
                a)  2x³ - px² + 6x - 3p is exactly divisible by x + 2
                b)  (x⁴ - p²x + 3 - p) / (x - 3) has a remainder of 4

Answers:
a) The remainder is 2(-2)³ - p(-2)² + 6(-2) - 3p = -16 -4p -12 - 3p = -28 - 7p = 0. Then p = -4
b) The remainder is 34 - p²(3) + 3 - p = 84 - 3p² - p = 4. Then 3p² + p - 80 = 0, (p - 5) (3p + 16) = 0 and p = 5, -16/3

4.  a) Show that x⁵ + a⁵ is exactly divisible by x + a?
     b) What is the remainder when y⁶ + a⁶ is divided by y + a? 

Answers:
a) P(x) = x⁵ + a⁵; then P(-a) = (-a)⁵ + a⁵ = -a⁵ + a⁵ = 0. Since P(-a) = 0, x⁵ + a⁵ is exactly divisible by x + a
b) P(y) = y⁶ + a⁶. Remainder = P(-a) = (-a)⁶ + a⁶ = a⁶ + a⁶ = 2a⁶


Hope this helps!! :) I'll end this blog with a video explaining why synthetic division works =)