Synthetic Division

Synthetic Division : This method is used to find remainder as well as used in factorization.

When you are dividing by a linear factor, you don't "have" to use long polynomial division; instead, you can use synthetic-division, which is much quicker.
Example : Find the remainder of x2 + 5x + 6 and x-1.
Write the coefficients ONLY inside an upside-down division symbol:
Make sure you leave room inside, underneath the row of coefficients, to write another row of numbers later. Put the test zero, x -1=0 ⇒x = 1, at the left:
Take the first number inside, representing the leading coefficient, and carry it down, unchanged, to below the division symbol:
Multiply this carry-down value by the test zero, and carry the result up into the next column:
Add down the column:
Multiply the previous carry-down value by the test zero, and carry the new result up into the last column:
Add down the column: This last carry-down value is the remainder.
So remainder is 12.
3) x3+7x2+7x -15

Solution :
As sum of all the coefficient ( 1 + 7 + 7 -15 )= 0 so (x -1 ) is one of the factor

After getting one factor find the other factors by Synthetic - Division.

As at the bottom of the line, there are 3 numbers so degree of the polynomial will be (3 -1 =2) and the numbers are 1,8,15.So the polynomial will be

x 2 + 8x + 15

Now find the factors of 15 in such a way that the addition and subtraction of that factors will be 8x.

So the factors of 15 = 5 and 3

x3+7x2+7x -15 = (x-1)(x + 5)(x + 3) are the factors.


Degree of the Polynomial
Zeros of Polynomial
Remainder Theorem
Find remainder by Synthetic Division
Rational root test in Polynomial
Solved Examples on Polynomial

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