Remainder and Factor Theorem

1
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
2
Intro to Polynomials
3
Intro to Polynomials
Simple Intro to Polynomials
http//www.glencoe.com/sec/math/algebra/algebra1/a
lgebra1_05/ brainpops/index.php4/na
More detailed Intro to Polynomials
http//www.youtube.com/watch?v18OFfTyic7g
4
Long Division of Polynomials
Simple Example
http//www.youtube.com/watch?vl6_ghhd7kwQ
More difficult example
http//www.youtube.com/watch?vFTRDPB1wR5Y
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Long Division of Polynomials

• Example 1

Dividend
Divisor
Quotient
In this case, the division is exact and Dividend
Divisor x Quotient
6
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
7

• 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
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Division by a Monomial
Divide
Rewrite
Divide each term separately
9
Division by a Binomial
Divide
Divide using long division
10
Division of Polynomials

• Division of polynomials is similar to a division
  sum using numbers.

Consider the division 10 2 5
Consider the division ( x2 x ) ( x 1 )
5
2
10
-
10
-
0
0


11
-
-
-
-
-
0
12

• When the division is not exact, there will be a
  remainder.

Consider the division 7 2
Consider (2x3 2x2 x) (x 1)
3
2
7
-
-
6
1
-
remainder
-1
remainder
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Example 1
-

-
Degree here is not smaller than divisors degree,
thus continue dividing
-
Degree here is less than divisors degree, thus
this is the remainder
14
Example 2
-

-
Degree here is less than divisors degree, thus
this is the remainder
15
Example 3
-

-
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Short Division of Polynomials
Examples
17
Synthetic Division of Polynomials
Preview Example the link from long division to
synthetic division
http//www.mindbites.com/lesson/931-int-algebra-sy
nthetic-division- with-polynomials
Examples how to perform synthetic division on
linear divisors (and the link to remainder
theorem)
http//www.youtube.com/watch?vbZoMz1Cy1T4 http/
/www.youtube.com/watch?vnefo9cUo-wg http//www.y
outube.com/watch?v4e9ugZCc4rw http//www.youtub
e.com/watch?v1jvjL9DtGC4
Extra how to perform synthetic division on
quadratic divisors
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Remainder and Factor Theorem
Introduction to Remainder Theorem
http//library.thinkquest.org/C0110248/algebra/rem
factintro.htm http//www.youtube.com/watch?vPJd2
6kdLxWw
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Remainder and Factor Theorem
Introduction to Factor Theorem
http//www.youtube.com/watch?vWyPXqe-KEm4feature
related
Use of Factor Theorem to solve polynomial
equations
http//www.youtube.com/watch?vnXFlAj7zBzofeature
related http//www.youtube.com/watch?vtBjSW365p
nofeaturerelated http//www.youtube.com/watch?v
7qcCOry8FoQfeaturerelated