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Notes 2.2: Division of Polynomials
Pre-Calc. for AP Prep.
Date: ______
Goals: To understand the connections between factoring, division, zeros and
x intercepts.
To determine the quotients and/or remainders of polynomial division through
long division, synthetic division, and the remainder theorem.
I. Polynomial Division by Factoring
A. Ex:
Divide:
4 x2  8x  5
2x 1
Quotient: ______________
Remainder: _______________
Benefit: It’s quick
Drawbacks: It only works if we know how to factor the dividend.
It only works if the divisor is a factor of the dividend.
II. Polynomial Division by Long Division
Ex. 1
2 x  1 4 x2  8x  5
Quotient: ____________
Remainder: ___________
Benefits: It works regardless of our ability to factor the dividend.
It gives us the quotient and remainder whether the divisor is a factor of
the dividend or not.
 1
a. For our consideration later…Given f ( x)  4 x 2  8 x  5 , find f    .
 2
Ex. 2:
Divide 3x4 – 10x2 + 9x – 4
by x – 1 using long division
Quotient _____________
Remainder ____________
From the results, we can determine that the expression
3x4 – 10x2 + 9x – 4
= _______________________________________________
. Also, we learn that if we wanted to, we could write the quotient
3x 4  10 x 2  9 x  4
as
x 1
a. For our consideration later…Given f ( x)  3x 4  10 x 2  9 x  4 , find f (1) .
III. Synthetic Division: An algorithm that uses just the coefficients to divide a
polynomial by a linear factor.
Ex.
Divide 3x4 – 50x2 + 9x – 4
by x – 4.
_______________________
*Notice that we still accounted
for the cubic term.
a. For our consideration later… Given f ( x)  3x 4  3x 2  9 x  4 , find f 4 .
b. Based on the results above, write 3x4 – 50x2 + 9x – 4
and a cubic factor.
as a product of a linear factor
Benefits: Quick
Works regardless of factorability or factors.
Drawbacks: Might not be obvious that it is based in good mathematics.
IV. Remainder Theorem
A. Given f ( x)  x 3  3x 2  8x  9 , find f (3) .
B.
Divide x3 – 3x2 + 8x + 9 by x – 3
using synthetic division. Name the
quotient and remainder. Quickly guess the remainder before we start.
Quotient: _____________
_______________________
Remainder: ____________
C. The Remainder Theorem (Formally) - _____________________________________
_____________________________________________________________________
_____________________________________________________________________
Application – By plugging in “associated zeros”, we can determine if a linear factor is a
factor of a given polynomial.
Ex: Find the remainder when x3 – 3x2 + 8x + 9 is divided by x – 3 using the
remainder theorem.
Benefits: Quickly finds remainder
Quickly identifies whether an expression has a certain linear factor.
Drawbacks: You don’t know quotient.
V. Mixed Applications
1)
Determine if 2x + 3 is a factor of 4x3 + 3x - 9
2)
Find the remaining roots to 4x4 – 4x3 – 25x2 + x + 6 if two of the roots are x = -2
and x = 3
3)
When a polynomial P(x) is divided by 2x + 1 the quotient is x2 – x + 4 and the
remainder is 3. Find P(x).
HW: p. 61 #2- 10 even, 15, 18, 20, 22, 26