Math 3
3-5 Synthetic Division
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1.
Name ___________________________
I can extend the relationship between standard and factored forms of polynomials.
I can perform and interpret the results of polynomial division and synthetic division.
Perform the following polynomial division. Write your answer in the form p  x   d  x   q  x   r  x 
x  4 x3  2 x 2  5 x  6
2. Is x  4 a factor of p( x)  x3  2 x 2  5 x  6? Explain.
On a seemingly unrelated note, imagine your teacher (me) told you to find p ( 4) without using your calculator.
Due to the exponents, I’m sure most of you would complain. Back in the day before calculators, many
mathematicians complained about the same thing because evaluating a polynomial directly was difficult due to
the exponents. Always looking for a shortcut, mathematicians were able to figure out a way around the
exponents. You already know what standard form and factored form of a polynomial; however, there is
another form called nested form. Nested form is designed to help evaluate polynomials without having to deal
with the powers of x. Look at the following:
3.
x3  2 x 2  5 x  6 can be factored into
   x  2  x  5 x  6 
How did I get from standard form to nested form?
4. Try evaluating p ( 4) using the nested form.
p ( x) 
   x  2  x  5 x  6 
5. While the nested form helps by getting rid of exponents, it is still a little confusing with all the nested
parenthesis. So a method called synthetic substitution was developed. Follow below:
-4
1
2
-5
-6
Where the numbers 1, 2, -5, -6 from?
Note the value of the answer to question (5). Look back to the division problem in question (1). What do
you notice? Huh . . . What does this tell us about the process of synthetic substitution??
Note your answer to question (4). How is that related to your work in question (5)? What is the relationship
between evaluating a polynomial at a given value and polynomial division?
6. Use what you learned in question (5) to perform the below division.
p  x   d  x   q  x   r  x  Write your answer in the form
a.
x  2 2 x3  3x 2  5 x  6
b. x  3 x3  6 x 2  30
There is one BIG restriction on the use of synthetic division. The divisor must be in the form of 1x  k. In
other words, the divisor must be a binomial of degree 1 and have a leading coefficient of 1. We can work
around the leading coefficient of 1 issue, but the divisor MUST be linear.
7. If you want to divide 4 x3  6 x 2  2 x  12 by 2 x 1 with synthetic division, then we need to write 2 x 1
with a leading coefficient of 1. How can we do that?
8. Now divide 4 x3  6 x 2  2 x  12 by 2 x 1 with synthetic division using your answer from question (7) as
the divisor. Keep in mind that we changed our divisor so we need to adjust our answer to account for that
in our final answer (we will work through that part together)
*Note – If the divisor is NOT linear, you MUST use long division.
9. Your turn! Divide 3x 4  20 x3  15 x 2  18 x  108 by 3x  2 with synthetic division.
10.
Perform the following division using the method you think is appropriate and most efficient. Write your
answer in the form p  x   d  x   q  x   r  x  . What does your remainder tell you about the
polynomial p  x  ?
10a.
x
10b.
2 x3  7 x 2  7 x  6
x2
11.
Using synthetic substitution, solve the following:
11a.
If f ( x)  2 x5  1x 4  2 x what does f  2  
3
 x 2  5 x  6    x  3
11b.
If f ( x)  6 x 4  3x 2  4 is 2 a zero for f ( x) ?
12.
Is ( x  3) a factor of 2 x 4  5 x 3  3 x 2  4
13.
Is (4 x  3) a factor of 4 x 4  3x3  8 x 2  3