Post on 20-Jan-2016
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Remainder and Factor Theorem
(1) Intro to Polynomials-degree-identities-division (long, short, synthetic)
(2) Remainder Theorem-finding remainders-special case Factor Theorem-factorise & solve cubic equations
Intro to Polynomials
Degree
Terms
Coefficient
Constant
Value
http://www.youtube.com/watch?v=18OFfTyic7g
More detailed Intro to Polynomials
Simple Intro to Polynomials
http://www.glencoe.com/sec/math/algebra/algebra1/algebra1_05/brainpops/index.php4/na
Intro to Polynomials
Long Division of Polynomials
http://www.youtube.com/watch?v=l6_ghhd7kwQ
http://www.youtube.com/watch?v=FTRDPB1wR5Y
Simple Example
More difficult example
Example 1:
326
Dividend Divisor Quotient
In this case, the division is exact and
Dividend = Divisor x Quotient
Long Division of Polynomials
Example 2: The number 7 when divided by 2 will not give an
exact answer. We say that the division is not exact.
[7 = (2 x 3) + remainder 1 ]
In this case, when the division is NOT exact,
Dividend = Divisor x Quotient + Remainder
Long Division of Polynomials
Definition of degree: For any algebraic expression, the highest
power of the unknown determines the degree.
For division of polynomials, we will stop dividing until the degree of the expression left is smaller than the divisor.
Algebraic Expression
Degree
2x + 1 1
x3 - 5x 3
-3x2 + x + 4 2
Division by a Monomial
Divide: 2245 2565812 xxxxx
Rewrite:5 4 2
2
12 8 5 6 5
2
x x x x
x
Divide each term separately:
222
2
2
4
2
5
2
5
2
6
2
5
2
8
2
12
xx
x
x
x
x
x
x
x
223
2
53
2
546
xxxx
Division by a Binomial
Divide: 1511710 34 xxxx
Divide using long division
10171015 234 xxxxx
Insert a place holder for the missing term x 2
Division of Polynomials Division of polynomials is similar to a division
sum using numbers.Consider the division 10 ÷ 2 = 5Consider the division 10 ÷ 2 = 5
10
2 5
10 0
Consider the division ( xConsider the division ( x2 2 + x ) ÷ ( x + + x ) ÷ ( x + 1 )1 )
xx 21x
)( 2 xx
x
0
--
)1(2 xxxx5210
2x x26x
xxx 23 21x
)( 23 xx
2x x
2(x )x-
-
0
0
xxxx 234 602x2
)2( 4x
3x 2
1
-
-
x3
)6( 2x
0 x)( x-
0))(1(2 223 xxxxxx )
2
13(262 324 xxxxxx
Example 1: )1()2( 23 xxxx Example 2: xxxx 2)62( 24
When the division is not exact, there will be a remainder.
Consider the division 7 ÷ 2 Consider the division 7 ÷ 2
72 3
6
1
Consider (2xConsider (2x3 3 + 2x+ 2x22 + x) ÷ (x + 1) + x) ÷ (x + 1)
xxx 23 221x)22( 23 xx
22x
- -
1)32(7 1)12)(1()22( 223 xxxxx
0 x
1
)1( x-
-1remainder
remainder
7 13
2 2
3 22(2 2 ) 1
(2 1)( 1) 1
x x xx
x x
x424x
x16
xxx 47 23 3x
)3( 23 xx
2x x4
)124( 2 xx -
-
-
Example 1: )3()47( 23 xxxx
Degree here is not smaller than divisor’s degree, thus continue dividing
16
)4816( x
48Degree here is less than divisor’s degree, thus this is the remainder
3
48)164( 2
x
xx
)3()47( 23 xxxx
8x12
4
875 2 xx1x
)55( 2 xx
x5 12
)1212( x-
-
Example 2: )1()875( 2 xxx
Degree here is less than divisor’s degree, thus this is the remainder
1
4)125(
x
x
)1()875( 2 xxx
1x2
0
156 2 xx12 x
)36( 2 xx
x3 1
)12( x-
-
Example 3: )12()156( 2 xxx
13 x
)12()156( 2 xxx
‘Short’ Division of Polynomials
Examples
2 2
2
2
2
2
2
2 3 5 1
2 3 5
1
2 1 3 3
1
3 32
1
x x x
x x
x
x x
x
x
x
3 2
2
2 2
2
2
2 3 4
2 5
2 ( 2 5) 7( 2 5) 3 31
2 53 31
2 72 5
x x x
x x
x x x x x x
x xx
xx x
Synthetic Division of Polynomials
http://www.youtube.com/watch?v=bZoMz1Cy1T4
http://www.youtube.com/watch?v=nefo9cUo-wg
http://www.youtube.com/watch?v=4e9ugZCc4rw
*http://www.youtube.com/watch?v=1jvjL9DtGC4
Preview Example: the link from long division to synthetic division
http://www.mindbites.com/lesson/931-int-algebra-synthetic-division-with-polynomials
Examples: how to perform synthetic division on linear divisors (and the link to remainder theorem)
Extra: how to perform synthetic division on quadratic divisors
Remainder and Factor Theorem
Introduction to Remainder Theorem
http://library.thinkquest.org/C0110248/algebra/remfactintro.htm
http://www.youtube.com/watch?v=PJd26kdLxWw
Remainder and Factor Theorem
Introduction to Factor Theorem
http://www.youtube.com/watch?v=WyPXqe-KEm4&feature=related
Use of Factor Theorem to solve polynomial equations
http://www.youtube.com/watch?v=nXFlAj7zBzo&feature=related
http://www.youtube.com/watch?v=tBjSW365pno&feature=related
http://www.youtube.com/watch?v=7qcCOry8FoQ&feature=related