Chapter3 Polynomials

8/7/2019 Chapter3 Polynomials

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POLYNOMIALSPOLYNOMIALS

CHAPTER 3CHAPTER 3DCT1043DCT1043


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CONTENT 3.1 Polynomial Identities

3.2 Remainder Theorem,Factor Theorem and

Zeros of Polynomial

3.3 Partial Fractions Decomposition


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3.1:3.1:

POLYNOMIALSPOLYNOMIALSIDENTITIESIDENTITIES


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Objectives By 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 & multiplicationof polynomials

Perform division of polynomials


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Introduction Basic Vocabulary

Monomial

where a is a constant (coefficient) and the powern is

a non-negative integer called degree

Nonmonomial

Binomial

The sum or difference of 2 monomial having different

degrees

Trinomial

The sum or difference of 3 monomial having differentdegrees

Polynomial

The sum of monomials

nax


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Polynomial Identities

Polynomials

A word derived from Greek many terms

Polynomial in the variablexin algebraic expression

P

olynomial is write in the formwhere is a constants (coefficients), is the leading

coefficient and thepowern is a non-negative integer called the

degree of the polynomial (The highest power ofx)

Identity

2 equation which have the same solution though expressed

differently ( use (equivalent) sign)

for all values ofP x Q x P x Q x x| !

|

1

1 1 0...

n n

n n

a x a x a x a

1 1 0, ,..., ,n na a a a na


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Example1 (Identity)

1. Given that

for all values ofx.

Find the value ofa, b and c.

2. Given that

for all values ofx.

Find the value ofa and b.

23 22 5 1 1 x x ax x b x c !

3 22 6 5 2 2 2 x x x x ax b |

TIPS: To find unknowns in an identity,

a) Substitute suitable values ofx, or b) Equate coefficients of like powers ofx


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Adding & Subtracting Polynomials

Horizontal Addition & Subtraction

Group the like terms (monomials with the

same power) and then combine them

Vertical Addition & Subtraction

Vertically line up the like terms in each

polynomial and then add or subtract the

coefficients.


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Example 2 (Adding & Subtracting)

4 2 31 1 25 1

5 3 3

x x x x

3 2 312 9 11 3 4 5 8 x x x x x

5 4 2 4 30.003 1.89 5.5 0.33 1 1.3 2 1.556 x x x x x x

3 2 3 29 7 5 3 13 2 8 6 x x x x x x

3 2 3 22 5 9 7 5 3 x x x x x

A

B

C

D

E


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Multiplying Polynomials

Horizontal Multiplication

Using the Laws of exponents, & the

Commutative, Associative and Distributive

properties and then combine them

Vertical Multiplication

Write the polynomial with the greatest

number of terms in the top row.


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Example 3 (Horizontal Multiplication)

2 33 4 5 2 x x x

2 22 4x x

2 32 3 2 5 2 x x x x

3

5x

542y

A

B

C

D

E


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Example 4 (Vertical Multiplication)

2 3 23 2 2 5 4 x x x x x

2 4 5 2 3 x x x

2 3 22 3 5 4 7 x x x x x

A

B

C


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Special Product Formulas

2 2

2 2 2

2 2 2

3

Difference of two square

Perfect square 2

2

Perfect Cubes

x a x a x a

x a x ax a

x a x ax a

x a

!

!

!

3 2 2 3

3 3 2 2 3

3 3 2 2

3 3 2 2

3 3

3 3

Sum of two cubes

Difference of two cubes

x ax a x a

x a x ax a x a

x a x a x ax a

x a x a x ax a

!

!

!

!


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Example 5 (Special Product)

33 2x

3

2 5x

3 32 3 2 3a a

2

3 7x

25 6y

A

B

C

D

E


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Factoring Polynomials

Any polynomial

Look for common monomial factors

Binomials of degree 2 or higher

Check for a special products

Trinomials of degree 2

Check for a perfect square

Three or more terms Grouping factored out the common factor

from each of several groups of terms.


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Factoring Strategy

1. Factor out the Greatest Common Factors

(GCF)

2. Check for any Special Products 2 term or3 term

3. If not a perfect square use try and error or

grouping

4. See if any factors can be factored further


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Example 6 (Factoring any polynomial)

2 3 5 3 x x x

5 4 316 12 4 x x x

3 24 3 12 x x x

3 23 15 42 x x x

3 218 27x xA

B

C

D

E

4 3 21

5 26 7 y y y

23 10x x

28 22 5x x

26 19 7x x

23 20 28x x F

G

H

I

J


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Example 7 (Factoring special forms)

31

2 3x x

109 16x

4 81x

281 49x

2 9x A

B

C

D

E

327 y

3 8x

364 125x

216 56 49x x

216 64x x F

G

H

I

J


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Division of a Polynomial

RECALL DividingTwo integers

quotient146

divisor 7 1025 divident

remainder3

n

p n

n

M

dividend = divisor quotient + remainderv

Long Division for

polynomials

-The process is similar

like division for integers

-The process is stop

when the degree of the

remainder is less than the

degree of divisor


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Example 8 (Long Division)

3 3 3 x x x z

3 22 3 3 x x x z

5 4 3 22 5 2 2 2 3 3 x x x x x x z

3 23 4 7 1 x x x x z

4 3 23 2 5 1 x x x x x z

A

B

C

D

E


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3.2:3.2:

REMAINDER THEOREM,REMAINDER THEOREM,

FACTOR THEOREMFACTOR THEOREM& ZEROS OF POLYNOMIALS& ZEROS OF POLYNOMIALS


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Objectives At the end of this topic, you should be able to

Use the remainder and factor theorem

Identify the value ofa such that (x + a) is a factor ofP(x)

and factorizeP(x) completely

Find the roots and the zeros of a polynomial

Determine the complex zeros of a polynomial up to

degree three


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Remainder Theorem

The remainder theorem state that,

Ifa polynomial ( ) is divided by a linear divisor the remainder is ( )

( )

f x x a f a

f x x a Q x f a

!

If a polynomial ( ) is divided by a linear divisor the remainder is

( )

bf x ax b f

a

bf x ax b x f

a

!


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Example 9 (Remainder Theorem)

2 4x

2x

2 1x

3 24 5x x

3x A

B

C

D

Find the remainder if is divided by


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Example 10 (Remainder Theorem)

6a b !

3x

24 7 x px (a) The expression leaves a remainder

-2 when divided by

Find the value ofp

1x

3 22 x ax bx c (b) Given that the expression

leaves the same remainder when divided by

or by .

Prove that

2x


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Factor Theorem

1.If is a factor of the polynomial then 0

2.If 0 then is a factor of the polynomial

x a f x f a

f a x a f x

!

!

The factor theorem state that,

Means that,

is a factor of the polynomial 0x a f x f a !


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Example 11 (Factor Theorem)

1x

4 3 2

3 3 2 x x x x

(a) Determine whether or not is a factor of

the following polynomials.

6 2 1 4 x x x i)ii)

1x 3x (b) Determine whether or not and is a

factor of 3 22 2 3f x x x x!


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Zeros of Polynomial

Use to solve polynomial function

A zero (root) of a function fis any value ofx for

whichf(x) = 0

Number of real zeros

A polynomial function cannot have more

real zeros than its degree

The maximum number are n


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Descartes Rule of Signs

Let fdenote a polynomial function written instandard form.

The number ofpositive real zeros offeither equals thenumber of variation in the sign of the nonzerocoefficients off(x) or else equals that number less aneven integer (2)

The number ofnegative real zeros offeither equals thenumber of variation in the sign of the nonzerocoefficients off(-x) or else equals that number less aneven integer (2)


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Example 12 (Descartes Rule of Signs)

6 4 3 23 4 3 2 3f x x x x x x!

Determine the number ofmaximum real zeros,

positive real zeros and negative real zeros from

the following polynomials.

3 22 11 7 6f x x x x!

i)

ii)


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Rational Zeros Theorem

Let fbe a polynomial function of degree 1 or

higher of the form

where each coefficient is an integer.

If in lowest terms, is a rational zero off,then

p must be a factor of and q must be a factorof

11 1 0 0... , 0, 0n n

n n nf x a x a x a x a a a

! { {

p

q

na0a


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Example 13 (Rational Zeros Theorem)

5 4 3 25 12 24 32 16f x x x x x x!

Listing all the potential real zeros from the following

polynomials.

3 22 11 7 6f x x x x! i)

ii)


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Finding the Real Zeros

Step 1: Determine the maximum number of zeros degree

Step 2: Determine the number of positive & negative zeros Descartes Rule of Signs

Step 3: Identify those rational numbers that potentially canbe zeros Rational Zeros Theorem

Step 4: Test each potential rational zeros long division

Step 5: Repeat Step 3 if a zero is found

Step 6: If possible, use the factoring techniques to find thezeros


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Example 14 (Finding Real Zeros)

5 4 3 25 12 24 32 16f x x x x x x!

Find all the real zeros from the following

polynomials.

3 22 11 7 6f x x x x! i)

ii)


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3.3:3.3:

PARTIAL FRACTIONPARTIAL FRACTION


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Objectives

By 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

A linear factor

A repeated linear factor

A Quadratic factor that cannot be factorized A repeated quadratic factor


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What is partial fractions

Consider the problem of adding 2 fraction

The reverse procedure

2

3 2 5 1

43

12

x

x x x x

!

2

5 1 3 2

12

43

x

x x x x

!

Partial fraction decomposition Partial fraction

Partial fraction


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What is partial fractions

Any rational function

where the degree ofPis less than the degree of ,could be

expressed as a sum of relatively simpler rational functions,calledpartial fractions.

Iff(x) is improper (degree of is less than the degree of

P), then by long division, dividingP by until aremainderR (x) is obtained such that degree ofR is less

than the degree of .

P xf x

x!


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Case 1: The denominatorQ(x) is

a product of distinct linear factor

1 1 2 2

1 2

1 12 2

If ,n n

n

n n

Q x a x b a x b a x b

P x AA A

Q x a x b a x b a x b

!

!

K

K

Examples

3

2

2 2

2 3 2

1 31. 3. 5.

2 3 4 1

5 2 12. 4. 6.

1 2 4 2 3 2

x x x

x x x x

x x x x

x x x x x x


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Case 2: The denominatorQ(x) is a

product of repeated linear factors

1 2

2

If ,n

n

n

Q x ax b

P x AA A

Q x ax b ax b ax b

!

!

K

Examples

2 2

3

2 32

1 3 41. 3.

2 2 1

12. 4.

1 2 1

x

x x x

x x x

x x x x


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Case 3: The denominatorQ(x)

contains irreducible quadratic factors

2

2

If ,Q x a x bx c

P x Ax B

Q x a x bx c

!

!

Examples

2

32

2

22

2 41. 3.

45

7 4 4 3 22. 4.

4 4 32 5

x x x

x xx

x x x

x xx x


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Case 4: Q(x) contains a repeated

irreducible quadratic factors

2

1 1

2 2

If ,n

n n

n

x ax bx c

P x xx

x ax bx cax bx c

!

!

K

Examples

2

2 22 2

3

1. 2.5 1 1

x x

x x x x


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Chapter3 Polynomials