2.5. Polynomial Long Division and Synthetic Division http://www.ck12.org
Second, check your result with the Remainder Theorem which states that the original function evaluated at 3 must
be 37. Notice the notation indicating to substitute 3 in forx.
(x^3 + 2 x^2 − 5 x+ 7 )|x= 3 = 33 + 2 · 32 − 5 · 3 + 7 = 27 + 18 − 15 + 7 = 37
Example C
Use Synthetic Division to divide the same rational expression as the previous example.
Solution: Synthetic division is mostly used when the leading coefficients of the numerator and denominator are
equal to 1 and the divisor is a first degree binomial.
x^3 + 2 xx^2 −− 35 x+ 7
Instead of continually writing and rewriting thexsymbols, synthetic division relies on an ordered spacing.
- 3 )1 2−5 7
Notice how only the coefficients for the denominator are used and the divisor includes a positive three rather than
a negative three. The first coefficient is brought down and then multiplied by the three to produce the value which
goes beneath the 2.
+ 3 )1 2−5 7
↓ 3
1
Next the new column is added. 2+ 3 =5, which goes beneath the 2ndcolumn. Now, multiply 5·+ 3 =15, which
goes underneath the -5 in the 3rdcolumn. And the process repeats...
+ 3 )1 2−5 7
↓ 3 15 30
1 5 10 37
The last number, 37, is the remainder. The three other numbers represent the quadratic that is identical to the
solution to Example B.
1 x^2 + 5 x+ 10
Concept Problem Revisited
Identifying roots of polynomials by hand can be tricky business. The best way to identify roots is to use the rational
root theorem to quickly identify likely candidates for solutions and then use synthetic or polynomial long division
to quickly and effectively test them to see if their remainders are truly zero.
Vocabulary
Polynomial long divisionis a procedure with rules identical to regular long division. The only difference is the
dividend and divisor are polynomials.
Synthetic divisionis an abbreviated version of polynomial long division where only coefficients are used.
Guided Practice
- Divide the following polynomials.