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Dividing Polynomials
March 07, 2014
Basic Method for Solving polynomials of the type:
ax3 + bx2 + cx + d = 0
&
ax4 + bx3 + cx2 + dx + e = 0
1. find linear factors using graphing calculator/Theorems
2. Divide polynomial by linear factor using:
3. solve remaining quadratic
long division
or
synthetic division
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Dividing Polynomials
March 07, 2014
Using Division to find Factors
Long division a. similar to long division of 2 & 3 digit numbers
b. must include "0 terms" for missing terms
c. factors are the divisor & the quotient if remainder = 0
d. subtraction is used in long division
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Dividing Polynomials
March 07, 2014
Synthetic division
a. simplified method of dividing polynomials which omits the variables & exponents
b. the divisor number 'a' comes from (x ­ a) (think of it as the value of x from x ­ a = 0) reverse sign of 'a'
c. must include "0" for coefficient of missing terms
d. Addition is used throughout this process
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Dividing Polynomials
March 07, 2014
Divide
by
Linear
(x -
using long division
Pull
Pull
divisor
1. divide 1
"Think
2. Align t
Pull
3. multip
Pull
4. bring d
Pull
1.
Pull
Dividing a polynomial by a linear factor
Rema
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Dividing Polynomials
March 07, 2014
Divide
Linear factor
(x - a)
using long division
by
divisor
Pull
Pull
Pull
Subtract (changes to add & changes signs
1. divide 1st term by "x" in (x - a)
"Think: what times "x" is "x2"?
2. Align terms on top by degree
3. multiply, then subtract each piece
Pull
4. bring down next term, & repeat
Pull
1.
Pull
Dividing a polynomial by a linear factor
Remainder
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Dividing Polynomials
March 07, 2014
Dividing a polynomial by a linear factor using
Divide
by
Linea
(x
using long division
divisor
Pull
2.
Pull
LONG DIVISION
Rema
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Dividing Polynomials
March 07, 2014
Dividing a polynomial by a linear factor using
Divide
by
Linear factor
(x - a)
using long division
divisor
Pull
2.
Pull
LONG DIVISION
Remainder
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Dividing Polynomials
March 07, 2014
3.
must be standard form,
if not, include 0x2, 0x...
for any missing terms
4.
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Dividing Polynomials
March 07, 2014
3.
must be standard form,
if not, include 0x2, 0x...
for any missing terms
4.
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Dividing Polynomials
March 07, 2014
SYNTHETIC DIVISION
Linear factor
(x - a)
*use the number part only
*reverse the sign, to create the "zero"
For the
Pull
Pull
For the
Polynomial
*use only coefficients
*include signs
*fill in missing terms with "0's"
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Dividing Polynomials
1.
March 07, 2014
Use synthetic division to divide by
Pull
1. write
& reve
Pull
2. write
use "
Pull
3. Bring
Pull
4. Multi
Pull
5. repe
mult
e Active Math
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Dividing Polynomials
1.
March 07, 2014
Use synthetic division to divide by
Pull
1. write number from (x - a)
& reverse sign in
Pull
2. write the coefficients
use "0" for missing terms
Pull
3. Bring down 1st coefficient
Pull
4. Multiply then add
did not divide evenly but we do have a function value
Pull
5. repeat with new number
multiply & add
see Active Math
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Dividing Polynomials
March 07, 2014
Divide using synthetic division
2.
3.
4.
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Dividing Polynomials
March 07, 2014
Divide using synthetic division
2.
X+1 is factor, x = -1 is zero
Divides
Evenly
x2 + 6x + 9 Depressed Polynomial
3.
1 degree less than
original polynomial
Does not Divide Evenly
4.
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Dividing Polynomials
March 07, 2014
Example for finding a function value using synthetic division & the Remainder Theorem
Use synthetic division to find P(­2)
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