1. Polynomial Synthetic Division
The shortcut to long division

2. Remember…
𝑝(𝑥)= 𝑥 2 −11𝑥+30
𝑑(𝑥)=𝑥−3
Find the quotient q(x) and the remainder r(x) if the first polynomial p(x) is divided by the second polynomial d(x). Then write in the form p(x)=q(x)d(x)+r(x):

3. Remember…
𝑝(𝑥)= 𝑥 2 −11𝑥+30
𝑑(𝑥)=𝑥−3
Find the quotient q(x) and the remainder r(x) if the first polynomial p(x) is divided by the second polynomial d(x). Then write in the form p(x)=q(x)d(x)+r(x): x -8 -3 x2
-11x
+30
-x2
+3x ↓
-8x 8x
-24 6

4. Synthetic Division
For synthetic division we use only the numbers (the coefficients and constant) from our polynomial function. Our polynomial must:
be in descending order – largest exponent to smallest exponent
have a term for every exponent after the first term
If there is a missing term, we add a new one in the form of 0xn
We get our divisor “a” from either our factor (x-a) or our zero x=a
If “a” is a factor of the polynomial, the remainder will ALWAYS be 0.
We can ONLY use synthetic division when we are dividing by something in the form (x-a)

5. Forms of an Answer 𝑝(𝑥 𝑑(𝑥 =𝑞(𝑥)+ 𝑟(𝑥) d(𝑥) 𝑝(𝑥)=𝑞(𝑥)𝑑(𝑥)+𝑟(𝑥
Depending on what we are being asked our answer can take many forms. We have the same two forms as yesterday, as well as a completely factored form. Remember: p(x) is our original polynomial, d(x) is our divisor, q(x) is our quotient, r(x) is our remainder.
𝑝(𝑥 𝑑(𝑥 =𝑞(𝑥)+ 𝑟(𝑥) d(𝑥)
𝑝(𝑥)=𝑞(𝑥)𝑑(𝑥)+𝑟(𝑥
𝑝(𝑥)=(𝑓𝑎𝑐𝑡𝑜 𝑟 𝑛 )(𝑓𝑎𝑐𝑡𝑜 𝑟 𝑛−1 )...(𝑓𝑎𝑐𝑡𝑜 𝑟 1 )(𝑓𝑎𝑐𝑡𝑜 𝑟 0

6. Now with Synthetic Division
𝑝(𝑥)= 𝑥 2 −11𝑥+30
𝑑(𝑥)=𝑥−3
Find the quotient q(x) and the remainder r(x) if the first polynomial p(x) is divided by the second polynomial d(x). Then write in the form p(x)=q(x)d(x)+r(x):

7. Now with Synthetic Division
𝑝(𝑥)= 𝑥 2 −11𝑥+30
𝑑(𝑥)=𝑥−3
Find the quotient q(x) and the remainder r(x) if the first polynomial p(x) is divided by the second polynomial d(x). Then write in the form p(x)=q(x)d(x)+r(x): 3 1
-11 30 ↓
-24 -8 6

8. Example #2
𝑝(𝑥)=3 𝑥 6 +2 𝑥 3 −176
𝑎=−2
Divide p(x) by the zero given and write in factored form, that is, write p in the form p(x)=(x-a)q(x)

9. Example #2
𝑝(𝑥)=3 𝑥 6 +2 𝑥 3 −176
𝑎=−2
Divide p(x) by the zero given and write in factored form, that is, write p in the form p(x)=(x-a)q(x) -2 3 2
-176 ↓ -6 12
-24 44
-88
176
-22

10. Example #3
Factor Completely Using Synthetic Division
𝑥= 1 2
𝑥=−3

11. Example #3 Factor Completely Using Synthetic Division -3 2 5 20 -12 ↓
𝑥= 1 2
𝑥=−3 -3 2 5 20
-12 ↓ -6 3
-24 12 -1 8 -4 1 4

12. Assignment
Polynomial Synthetic Division Worksheet