Try our Free Online Math Solver!

Polynomial Division, The Remainder Theorem, and The Factor Theorem
Polynomial Division , The
Remainder Theorem, and The
Factor Theorem
Long Division
Notice: If we get a remainder of zero the quotient is a factor of the dividend
We have known how to find factors of integers using division for years
We can use long division on polynomials in a like manner
Setup Put the divisor outside 
Put the dividend inside with
every degree of the variable shown
Warning: This book may represent the remainder as
some function
R(x) and say R(x) = (some constant.)
ONLY if we write the QUOTIENT as the sum of two separate
functions
would this be acceptable.
Otherwise the answer is as shown as
Synthetic Division
• Can only use when the divisor has a degree of 1.
Otherwise a different method is needed.
• Uses the coefficients
• Must backfill for all powers of the variable
Setup Put the solution outside 

The 1st coefficient moves down 

Multiply by solution and move 
Add 
The numbers below the line are the coefficients to a term that is lower by 1 degree
GO
The remainder Theorem
• If a number c is substituted for x in the polynomial p(x)
• then the result p(c) is the remainder that would result from dividing p(x) by
(xc)
Find the remainder for: f(1) f(3) f(2)
Check with synthetic division
The Factor Theorem
• For a polynomial p(x)
• if p(c) = 0
• then (xc) is a factor of p(x)
Test the following numbers to see if they are zeros using
the table feature in the calculator
1, 2, 3
If they are zeros, give the factors
Prev  Next 