http://www.ck12.org Chapter 2. Polynomials and Rational Functions
Polynomial long division is identical to regular long division. Synthetic division is a condensed version of regular
long division where only the coefficients are kept track of. In Example B polynomial long division is used and in
Example C synthetic long division is used.
Example A
Identify all possible rational solutions of the following polynomial using the Rational Root Theorem.
12 x^18 − 91 x^17 +x^16 +···+ 2 x^2 − 14 x+ 5 = 0
Solution: The integer factors of 5 are 1, 5. The integer factors of 12 are 1, 2, 3, 4, 6 and 12. Since pairs of factors
could both be negative, remember to include±.
±qp=^11 ,^12 ,^13 ,^14 ,^16 , 121 , 15 ,^52 ,^53 ,^54 ,^56 , 125
While narrowing the possible solutions down to 24 possible rational answers may not seem like a big improvement,
it surely is. This is especially true considering there are only a handful of integer solutions. If this question required
you to find a solution, then the Rational Root Theorem would give you a great starting place.
Example B
Use Polynomial Long Division to divide:
x^3 + 2 xx^2 −− 35 x+ 7
Solution: First note that it is clear that 3 is not a root of the polynomial because of the Rational Root Theorem and
so there will definitely be a remainder. Start a polynomial long division question by writing the problem like a long
division problem with regular numbers:
x− 3 )x^3 + 2 x^2 − 5 x+ 7
Just like with regular numbers ask yourself “how many times doesxgo intox^3 ?” which in this case isx^2.
x^2
x− 3 )x^3 + 2 x^2 − 5 x+ 7
Now multiply thex^2 byx−3 and copy below. Remember to subtract the entire quantity.
x^2
x− 3 )x^3 + 2 x^2 − 5 x+ 7
−(x^3 − 3 x^2 )Combine the rows, bring down the next number and repeat.
x^2 + 5 x+ 10
x− 3 )x^3 + 2 x^2 − 5 x+ 7
−(x^3 − 3 x^2 )
5 x^2 − 5 x
−( 5 x^2 − 15 x)
10 x+ 7
−( 10 x− 30 )
37The number 37 is the remainder. There are two things to think about at this point. First, interpret in an equation:
x^3 + (^2) xx−^2 − 35 x+ (^7) = (x (^2) + 5 x+ 10 )+x (^37) − 3