Document1
White Plains High School
Mr. Stanton
SWBAT to express a polynomial using the Remainder Theorem
Do Now
1. Determine 𝑓(2) if 𝑓(𝑎) = 4𝑎3 − 7𝑎2 + 𝑎 − 17
2. Use synthetic division to find the remainder when a root of 2 is used
with 4𝑎2 − 7𝑎2 + 𝑎 − 17
3. Determine 𝑓(3) if 𝑓(𝑥) = 3𝑥 3 − 5𝑥 2 − 8𝑥 − 6.
4. Use synthetic division to find the remainder when a root of 3 is used
with 3𝑥 3 − 5𝑥 2 − 8𝑥 − 6.
5. What is the significance of this?
Pre-Calc Honors: The Division Algorithm and Remainder Theorem
DIVISION ALGORITHM FOR POLYNOMIALS: If P(x) and D( x)  0 are
polynomials, where the degree of D(x) ≤ P(x), then ! polynomials Q(x), R(x):
𝑃(𝑥)
𝑅(𝑥)
= 𝑄(𝑥) +
𝐷(𝑥)
𝐷(𝑥)
In other words,
OR
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑
𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟
= 𝑄𝑢𝑜𝑡𝑖𝑒𝑛𝑡 +
𝐷𝑖𝑣𝑖𝑠𝑜𝑟
𝐷𝑖𝑣𝑖𝑠𝑜𝑟
𝑃(𝑥) = 𝑄(𝑥) ∙ 𝐷(𝑥) + 𝑅(𝑥)
If 𝑅(𝑥) = 0, then we say that 𝐷(𝑥) divides evenly into 𝑃(𝑥), or even better, 𝐷(𝑥) is a
factor of 𝑃(𝑥).
REMAINDER THEOREM: When 𝐷(𝑥) is 𝑥 − 𝑐, the equation
𝑃(𝑥) = 𝑄(𝑥) ∙ 𝐷(𝑥) + 𝑅(𝑥) becomes 𝑃(𝑥) = (𝑥 − 𝑐) ∙ 𝑄(𝑥) + 𝑃(𝑐) and therefore
𝑃(𝑐) is the remainder when 𝑃(𝑥) is divided by 𝑥 − 𝑐.
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Document1
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White Plains High School
Mr. Stanton
Document1
White Plains High School
Mr. Stanton
Summary on Synthetic Division
Given a polynomial function P(x), explain how you would use synthetic division to…
a) Divide P(x) by (x – 3)
b) Determine if x = 3 is a root (solution) of polynomial P(x)
c) Determine if (x – 3) is a factor of polynomial P(x)
d) Determine if (2x – 3) is a factor of P(x)
e) Determine P(3)
f) Solve a polynomial equation given a root 𝑟
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Document1
White Plains High School
Mr. Stanton
In 11-16…
a) Determine if the 1st polynomial is a factor of the 2nd polynomial
b) Write the 2nd polynomial in the form P ( x)  Q ( x)  D ( x)  R
partial answers are in parentheses
11.
12.
13.
14.
15.
16.
D( x)  ( x  1); P( x)  x5  x 4  x3  x 2  x  1
D( x)  ( x  1); P( x)  x8  x5  2
D( x)  ( x  1); P( x)  x10  x9  x  1
D( x)  ( x  2); P( x)  x8  2 x 7  x  2
D( x)  ( x  2i); P( x)  x3  x 2  4 x  4
D( x)  ( x  i); P( x)  x3  x 2  x  1
(yes)
(no)
(yes)
(yes)
(yes, hint: conjugates!)
(yes)
17. Determine a value k such that the 1st polynomial is a factor of the 2nd polynomial
D( x)  ( x  2); P( x)  2 x3  3x 2  kx  k  1
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(k=3)