Chapter3 Polynomials
This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA
Overview
Download & View Chapter3 Polynomials as PDF for free.
More details
- Words: 2,558
- Pages: 43
Loading documents preview...
| |
O.1 Polynomial Identities Ô O.2 Remainder Theorem, Factor Theorem and Zeros of Polynomial Ô O.O Partial Fractions Decomposition Ô
[email protected]
i
|
Ô
y the end of this topic, you should be able to -
Recognize monomials, binomials & trinomials Define polynomials, & state the degree of a polynomial & the leading coefficient Perform addition, subtraction & multiplication of polynomials Perform division of polynomials
[email protected]
á
! " #$ % $!&$ ' ·
m · ! ü !
ü
V · ·
·
·
·
·
| ·
[email protected]
r
| &' ($& ·
| ·
·
ü
| o | ü ! ! ` ` !` ! ! ! ` !` !
! ü ü ü ·
·
·
ü ]
| > ] >
[email protected]
| > >
)$(*& * ' 1.
Given that ! ! > `
> `
for all values of . Find the value of !, and . 2.
Given that " ! > for all values of . Find the value of ! and .
>
!
| ü # # # $ ü
[email protected]
M
+ , ! $ + | &' ($& · aorizontal ·
Group the like terms *monomials with the same power and then combine them
· Vertical ·
Addition & Subtraction
Addition & Subtraction
Vertically line up the like terms in each polynomial and then add or subtract the coefficients.
[email protected]
ë
)$(*& - .
+ , ! $ +/
> %
#
! % ' ! > >
` % `` á > > ! &
§` á !
>
[email protected]
' !
>`
& "
` § ! ` ! `&% á !! > `
>`
á
`!!"
w
!&*&' + | &' ($& · aorizontal ·
Using the Laws of exponents, & the Commutative, Associative and Distributive properties and then combine them
· Vertical ·
Multiplication
Multiplication
Write the polynomial with the greatest number of terms in the top row.
[email protected]
`
)$(*& . 0 $& !&*&$ / >!
#
> ð
á !
>
> >á
>
á >
[email protected]
!
>
!
``
)$(*& .% $& !&*&$ /
>
#
>
>
á ! >
>
! á
'
> !
á
[email protected]
`i
*$& | ! 1 (!&$ ( ü > ! > ! ! | > !
> !
! !
| )# > !
! ! !
> !
! ! !
! !
ü# ! > ! > ! ! ( ü# ! > ! > [email protected]
! ! `
)$(*& 2 .*$& | !/ #
>
'
>! ð "
> !
>
>!
>!
[email protected]
`á
Factoring Polynomials · Any ·
polynomial
Look for common monomial factors
· inomials ·
of degree 2 or higher
Check for a special products
· Trinomials ·
Check for a perfect square
· Three ·
of degree 2
or more terms
Grouping ± factored out the common factor from each of several groups of terms.
[email protected]
`r
Factoring Strategy 1. 2.
Factor out the Greatest Common Factors *GCF Check for any Special Products ·
O. 4.
2 term or O term
If not a perfect square use µtry and error¶ or grouping See if any factors can be factored further
[email protected]
`
)$(*& 3 .1$ + $ ' * &' ($&/ 1
&
`! á
4
" `% '
`" ! ` á
`
>
`! ð á
á `
5
& !
`&
#
[email protected]
'
á
! >
" ð ' ð
`M
)$(*& 6 .1$ + *$& 7 ( /
%
1
`" "á
#
&` á%
4
`" !" á%
% `"`
&
`
' ð
á &`
5
"á `!
[email protected]
`ë
7 $ | &' ($& ·
RECALL ± Dividing Two integers
`á" O
' ` ! O
m O
+ 7 * &' ($& XThe process is similar like division for integers XThe process is stop when the degree of the remainder is less than the degree of divisor
* + [email protected]
`w
)$(*& 8 . + /
>
#
>
! > `
> >
>
>
á
á ' > `
> !
[email protected]
! á >
i
- 9 1 , : 1 |
Ô
## , - ! ! - - ^ - - ( - -
[email protected]
ii
($ ; ( ·
The remainder theorem state that,
£ # ! £
£ >
>
£
>
§ £ # ! £ ! § £ >! > £ !
[email protected]
i
)$(*& < . ($ ; (/ Find the remainder if
#
á
`
[email protected]
á !
is divided by
iá
)$(*& . ($ ; (/ *a The expression á ' leaves a remainder X2 when divided by Find the value of *b Given that the expression
!
leaves the same remainder when divided by
or by ` . Prove that [email protected]
" ir
Factor Theorem ·
The factor theorem state that,
> ·
£ >
£
>
Means that, ` > £
[email protected]
>
£ > £
>
>
£ >
i
)$(*& .1$ ; (/ *a Determine whether or not
` is a factor of
the following polynomials. i
ii
"
á
> ` á
*b Determine whether or not factor of£ >
[email protected]
` and is a
iM
: 7 | &' ($& · Use ·
to solve polynomial function
-
£ ü £ *
· Vumber
of real zeros
A polynomial function cannot have more real zeros than its degree · The maximum number are ·
[email protected]
ië
$ = !& 7 + ·
p £ ü
·
# - £ # - £ #
·
# - £ # - £ #
[email protected]
iw
)$(*& - . $ = !& 7 + / Determine the number of maximum real zeros, positive real zeros and negative real zeros from the following polynomials. i
£ > " á á
ii
£ > `` ' "
[email protected]
Rational Zeros Theorem ·
p £ # ` ` £ > ` ` ü
ü - £
# #
[email protected]
`
)$(*& . $ $& : ; (/ Listing all the potential real zeros from the following polynomials.
i ii
£ > `` ' " !
£ > !
[email protected]
á
` á
`"
i
1 + ; $& : ·
Step 1: Determine the maximum number of zeros ± degree
·
Step 2: Determine the number of positive & negative zeros ± Descartes¶ Rule of Signs
·
Step O: Identify those rational numbers that potentially can be zeros ± Rational Zeros Theorem
·
Step 4: Test each potential rational zeros ± long division
·
Step 5: Repeat Step O if a zero is found
·
Step 6: If possible, use the factoring techniques to find the zeros
[email protected]
)$(*& .1 + $& : / Find all the real zeros from the following polynomials.
i ii
£ > `` ' " !
£ > !
[email protected]
á
` á
`"
á
| 1
Ô
y the end of this topic, you should be able to Define partial fractions - Obtain partial fractions decomposition when the denominators are in the form of -
Ô Ô Ô Ô
A linear factor A repeated linear factor A Quadratic factor that cannot be factorized A repeated quadratic factor
[email protected]
±;$ *$ $& 7 $
·
Consider the problem of adding 2 fraction
á ·
! ` `
The reverse procedure
! ` ` á Partial fraction decomposition
[email protected]
Partial fraction Partial fraction
M
±;$ *$ $& 7 $
·
£ >
>
>
ü # ·
£ # # #
[email protected]
ë
$ ; ( $ *o $ * ! 7 & $ 7$ > > ` ` | > ` > > ` ` ·
> >
;
> ;
>
Examples
`
`
>
>
> ` >
[email protected]
á
! á á
! ` ` "
w
$ - ; ( $ *o $ * ! 7
*$ & $ 7$
> > | > ` > > ·
>
;
>
Examples
`
`
> [email protected]
> ` >
á
á > `
`
> `
á
$ ; ( $ *o $ !& >!$ $ 7$ > > | > > > ·
Examples
`
> !
' á > > ! [email protected]
á á
á á á á á`
$ *o $
$ *$ !& >!$ $ 7$ >
>
> ·
>!
>!
`
`
;
>!
Examples
`
> [email protected]
!
> ` >
`
ái
? @
O.1 Polynomial Identities Ô O.2 Remainder Theorem, Factor Theorem and Zeros of Polynomial Ô O.O Partial Fractions Decomposition Ô
[email protected]
i
|
Ô
y the end of this topic, you should be able to -
Recognize monomials, binomials & trinomials Define polynomials, & state the degree of a polynomial & the leading coefficient Perform addition, subtraction & multiplication of polynomials Perform division of polynomials
[email protected]
á
! " #$ % $!&$ ' ·
m · ! ü !
ü
V · ·
·
·
·
·
| ·
[email protected]
r
| &' ($& ·
| ·
·
ü
| o | ü ! ! ` ` !` ! ! ! ` !` !
! ü ü ü ·
·
·
ü ]
| > ] >
[email protected]
| > >
)$(*& * ' 1.
Given that ! ! > `
> `
for all values of . Find the value of !, and . 2.
Given that " ! > for all values of . Find the value of ! and .
>
!
| ü # # # $ ü
[email protected]
M
+ , ! $ + | &' ($& · aorizontal ·
Group the like terms *monomials with the same power and then combine them
· Vertical ·
Addition & Subtraction
Addition & Subtraction
Vertically line up the like terms in each polynomial and then add or subtract the coefficients.
[email protected]
ë
)$(*& - .
+ , ! $ +/
> %
#
! % ' ! > >
` % `` á > > ! &
§` á !
>
[email protected]
' !
>`
& "
` § ! ` ! `&% á !! > `
>`
á
`!!"
w
!&*&' + | &' ($& · aorizontal ·
Using the Laws of exponents, & the Commutative, Associative and Distributive properties and then combine them
· Vertical ·
Multiplication
Multiplication
Write the polynomial with the greatest number of terms in the top row.
[email protected]
`
)$(*& . 0 $& !&*&$ / >!
#
> ð
á !
>
> >á
>
á >
[email protected]
!
>
!
``
)$(*& .% $& !&*&$ /
>
#
>
>
á ! >
>
! á
'
> !
á
[email protected]
`i
*$& | ! 1 (!&$ ( ü > ! > ! ! | > !
> !
! !
| )# > !
! ! !
> !
! ! !
! !
ü# ! > ! > ! ! ( ü# ! > ! > [email protected]
! ! `
)$(*& 2 .*$& | !/ #
>
'
>! ð "
> !
>
>!
>!
[email protected]
`á
Factoring Polynomials · Any ·
polynomial
Look for common monomial factors
· inomials ·
of degree 2 or higher
Check for a special products
· Trinomials ·
Check for a perfect square
· Three ·
of degree 2
or more terms
Grouping ± factored out the common factor from each of several groups of terms.
[email protected]
`r
Factoring Strategy 1. 2.
Factor out the Greatest Common Factors *GCF Check for any Special Products ·
O. 4.
2 term or O term
If not a perfect square use µtry and error¶ or grouping See if any factors can be factored further
[email protected]
`
)$(*& 3 .1$ + $ ' * &' ($&/ 1
&
`! á
4
" `% '
`" ! ` á
`
>
`! ð á
á `
5
& !
`&
#
[email protected]
'
á
! >
" ð ' ð
`M
)$(*& 6 .1$ + *$& 7 ( /
%
1
`" "á
#
&` á%
4
`" !" á%
% `"`
&
`
' ð
á &`
5
"á `!
[email protected]
`ë
7 $ | &' ($& ·
RECALL ± Dividing Two integers
`á" O
' ` ! O
m O
+ 7 * &' ($& XThe process is similar like division for integers XThe process is stop when the degree of the remainder is less than the degree of divisor
* + [email protected]
`w
)$(*& 8 . + /
>
#
>
! > `
> >
>
>
á
á ' > `
> !
[email protected]
! á >
i
- 9 1 , : 1 |
Ô
## , - ! ! - - ^ - - ( - -
[email protected]
ii
($ ; ( ·
The remainder theorem state that,
£ # ! £
£ >
>
£
>
§ £ # ! £ ! § £ >! > £ !
[email protected]
i
)$(*& < . ($ ; (/ Find the remainder if
#
á
`
[email protected]
á !
is divided by
iá
)$(*& . ($ ; (/ *a The expression á ' leaves a remainder X2 when divided by Find the value of *b Given that the expression
!
leaves the same remainder when divided by
or by ` . Prove that [email protected]
" ir
Factor Theorem ·
The factor theorem state that,
> ·
£ >
£
>
Means that, ` > £
[email protected]
>
£ > £
>
>
£ >
i
)$(*& .1$ ; (/ *a Determine whether or not
` is a factor of
the following polynomials. i
ii
"
á
> ` á
*b Determine whether or not factor of£ >
[email protected]
` and is a
iM
: 7 | &' ($& · Use ·
to solve polynomial function
-
£ ü £ *
· Vumber
of real zeros
A polynomial function cannot have more real zeros than its degree · The maximum number are ·
[email protected]
ië
$ = !& 7 + ·
p £ ü
·
# - £ # - £ #
·
# - £ # - £ #
[email protected]
iw
)$(*& - . $ = !& 7 + / Determine the number of maximum real zeros, positive real zeros and negative real zeros from the following polynomials. i
£ > " á á
ii
£ > `` ' "
[email protected]
Rational Zeros Theorem ·
p £ # ` ` £ > ` ` ü
ü - £
# #
[email protected]
`
)$(*& . $ $& : ; (/ Listing all the potential real zeros from the following polynomials.
i ii
£ > `` ' " !
£ > !
[email protected]
á
` á
`"
i
1 + ; $& : ·
Step 1: Determine the maximum number of zeros ± degree
·
Step 2: Determine the number of positive & negative zeros ± Descartes¶ Rule of Signs
·
Step O: Identify those rational numbers that potentially can be zeros ± Rational Zeros Theorem
·
Step 4: Test each potential rational zeros ± long division
·
Step 5: Repeat Step O if a zero is found
·
Step 6: If possible, use the factoring techniques to find the zeros
[email protected]
)$(*& .1 + $& : / Find all the real zeros from the following polynomials.
i ii
£ > `` ' " !
£ > !
[email protected]
á
` á
`"
á
| 1
Ô
y the end of this topic, you should be able to Define partial fractions - Obtain partial fractions decomposition when the denominators are in the form of -
Ô Ô Ô Ô
A linear factor A repeated linear factor A Quadratic factor that cannot be factorized A repeated quadratic factor
[email protected]
±;$ *$ $& 7 $
·
Consider the problem of adding 2 fraction
á ·
! ` `
The reverse procedure
! ` ` á Partial fraction decomposition
[email protected]
Partial fraction Partial fraction
M
±;$ *$ $& 7 $
·
£ >
>
>
ü # ·
£ # # #
[email protected]
ë
$ ; ( $ *o $ * ! 7 & $ 7$ > > ` ` | > ` > > ` ` ·
> >
;
> ;
>
Examples
`
`
>
>
> ` >
[email protected]
á
! á á
! ` ` "
w
$ - ; ( $ *o $ * ! 7
*$ & $ 7$
> > | > ` > > ·
>
;
>
Examples
`
`
> [email protected]
> ` >
á
á > `
`
> `
á
$ ; ( $ *o $ !& >!$ $ 7$ > > | > > > ·
Examples
`
> !
' á > > ! [email protected]
á á
á á á á á`
$ *o $
$ *$ !& >!$ $ 7$ >
>
> ·
>!
>!
`
`
;
>!
Examples
`
> [email protected]
!
> ` >
`
ái
? @
Related Documents
Chapter3 Polynomials
January 2021 1
Polynomials
January 2021 0
Chapter3 Auditing 2
January 2021 1
Polynomials Unit Packet
January 2021 0
Text 8. Polynomials
January 2021 0
Chapter 14 Polynomials
March 2021 0More Documents from "Ozzy PingBoi"