College Algebra & Trigonometry, 2018a

124 CHAPTER 2. POLYNOMIAL AND RATIONAL FUNCTIONS
2.7 Synthetic Division
The process for polynomial long division (like the process for numerical long
division) has been separated somewhat from its logical underpinnings for a more
efficient method to arrive at an answer. For particular types of polynomial long
division, we can even take this abstraction one step further. Synthetic Division is
a handy shortcut for polynomial long division problems in which we are dividing
by a linear polynomial. This means that the highest power of x we are dividing by
needs to be x 1 . This limits the usefulness of Synthetic Division, but it will serve
us well for certain purposes. Let’s examine where the coefficients in our answer
come from when we divide by a linear polynomial:
2x 3
x − 5 ) 2x 4 − 6x 3 − 23x 2 +16x − 5
− 2x 4 +10x 3
4x 3 − 23x 2
Notice that the first coefficient in the answer is the same as the first coefficient in
the polynomial we’re dividing into. This is because we’re dividing by a polynomial
in the form 1x 1 −a. This also makes the power of the first term in the answer
one less than the power of the polynomial we are dividing into. Let’s see where
the subsequent coefficients in the answer come from:
2x 3 +4x 2
x − 5 ) 2x 4 − 6x 3 − 23x 2 +16x − 5
− 2x 4 +10x 3
4x 3 − 23x 2
− 4x 3 +20x 2
− 3x 2 +16x
The next coefficient in the answer (4) comes from the combination of the −6 and
the +10. The +10 came from multiplying the 2 in the answer by the 5 in the
divisor x − 5. The next coefficient in the answer will be −3, which comes from
multiplying the 4 (in the answer) by the 5 (in the divisor) and combining it with
the −23 in the polynomial we’re dividing into: