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