ee004 fhs lnt 003 chapter 1 jan11

8/22/2019 EE004 Fhs Lnt 003 Chapter 1 Jan11

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Learning Outcome:

1. Understand the definition of polynomials2. Add and subtract polynomials

3. Multiply polynomial

4. Use special products rule in polynomial multiplication

5. Use long division to divide polynomial

6. Solve the quadratic equations with FactorizationMethod, Completing the Square and Quadratic Formula


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Definition A polynomial is a single term or the sum of two or more

terms containing variables with whole number exponents.

A polynomial in the variablex is an algebraic expressionof the form:

anxn + an-1x

n-1+a1x + a0x0

where an 0; n is a nonnegative integer and coefficients a0,a1, a2 an are constant polynomial.x represents a real

number. Rules: Variables should have only nonnegativeand non-fraction integralexponentsand no variable isdenominator


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Classifying Polynomials

Monomiala polynomial with exactly one term

Binomiala polynomial that has two terms

Trinomiala polynomial with three terms


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Example 1 is a polynomial.

is not a polynomial because the second term

involves the division by the variable ofx

. is not a polynomial because the third

term contains an exponent which is not an non-negative integer.


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Exercise 1Determine whether the following statement is a

polynomial or not.

I. x2 +x1/2 + 9

II. x34x2 +

III. x-2 + 6x + 4

IV. 4x4

+ 2x + 7V. 20x3


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Standard form and degree A polynomials is in standard form if its terms are

arranged so that the powers of the variables are indescending orascending order

anxn + an-1x

n-1+a1x + a0a0 + a1x+ an-1x

n-1 + anxn

Degree of a polynomial equation is referred to the largestpower of the variable.


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Exercise 2Polynomial Degree Standard Form

x3 - 3x + 12

x10 -x208x

32 +x6

7x4


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Adding and Subtracting Polynomials are added and subtracted by combining like

terms

(ax + b) + (cx + d) = (a + c)x + (b + d) Following is the steps of solving the addition and

subtraction of polynomials

1. Rearrange like terms in the polynomials.

2. Add or subtract the like terms.


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Example 2


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Example 3


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Exercise 3 Solve the following problems, then write the

resulting polynomial in standard form.

1. (-6x3 + 5x2 - 8x + 9) + (17x3 + 2x2 - 4x -13)

2. (5x2 + 7x - 9) - (3x3 - 8) - (-x2 - 6x + 3)

3. (5x4 - 5x) + (8x - 7) - (2x2 - 3x - 9)

4. (6x4 + 2x) - (3x + 3x4)5. (x3 - 2x2 + 3x + 4) + (5 + 6x - 7x2 - 8x3)


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Multiplication of Polynomial In multiplication of polynomials, use the distributive

property to multiply each term by term, combine the liketerms and write the result in simplest form.

E.g:

ax(bx + c) = ax(bx) + ax(c)

If the multiplication operation involves the exponentfunctions, we have to add the powers of the polynomialfor multiplication.

E.g:

(ax2)bx5 = (ab)x2+5 = abx7


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Multiplication of Polynomial Use the distributive property to multiply the trinomial

by each term of the binomial. Multiply the monomials,then combine like terms.

(ax2 - bx + c)(dx- e)

= dx(ax2 - bx + c) -e(ax2 - bx + c)

= adx3 - bdx2 + dcx - aex2 + bex - ce

= adx3 + (- bd - ae)x2 + (dc+be)x - ce


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Example 4 Find the product of the following problem

Solution:


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Special Product Rules Certain types of products are important to deserve special

attention. These are stated in the following list of specialproduct rules. Each of them can be verified by directmultiplication.

Square of Binomial:(a + b)2 = a2 + 2ab + b2 Binomial Sum(a - b)2 = a22ab +b2 Binomial Difference

Sum and Difference of Two Squares:

(a - b)(a + b) = a2

b2

Cube of Binomial:(a + b)3 = a3 + 3a2b + 3ab2 + b3(a - b)3 = a33a2b + 3ab2b3


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Exercise 4

Find each product

1. (x + 4y)(3x - 5y)

2. (5x + 3y)2

3. (x + 1)(x2 -x +1)

4. (3x + 4)3

5. (2x + 4)(5x2 - 1)

6. (x - 1)(x - 3)

7. (2x)(x

2

+ 2x - 5)8. (6x + 5)(7x - 2)2

9. (5x - 3)3

10. (9x + 2y)(9x - 2y)


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Division of PolynomialsPolynomials division is one of the basic tools needed in the

study of the theory of polynomials. This subtopic willreviews on long division polynomials.

Long Division of PolynomialsIn order to perform the long division operation, the degreeof polynomials should be checked to be arranged indescending order.


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Example 5

Divide byand then write in the forms

Divisor

Remainderquotient

Divisor

Dividend


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Solution4y2 - 3y + 2 ) 8y3 - 18y2 + 11y - 6

Divisor

Dividend

- (8y3 - 6y2 + 4y)

-12y2 + 7y - 6-(-12y2 + 9y - 6)

-2y

2y Quotient

Remainder

234

232

234

61118822

23

yy

yy

yy

yyySo,

- 3


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Exercise 5Find the Quotient and Remainder of the following:

1. (x5 + 4x2 + 2x4 - 5x - 3x3 + 2) (x1 )

2. (2x3

3x2

+ 4x5)

(x2)3. (3x3x + 84x2 ) (4 + x )

4. (x5 + 10x2 + 4x37x4x5 ) (x1)

5. (4x339x28x ) (x3)

6. (x34x + 82x2 ) (2 + x)


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Quadratic Polynomial An expression of the form , where a, b and c

are constants with , is called a quadratic expression.

An equation of the form , where a, b and c

are constants with , is called a quadratic equation.

Quadratic equations can be solved by three methods,namely,

a) Factorization

b) Completing the square

c) Quadratic formula

cbxax 2

0a

02 cbxax0a


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Factorization

Example:

Solve the quadratic equations below by using factorization

method.

Solution:

x =0 ; 2x + 5 = 0

x =

baxxbxax 2

0522

xx

052

0522

xx

xx

2

5


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FactorizationIf

Note:

Ifc is positive, thenp and q must both have the same sign. Ifc is negative, thenp and q have opposite signs.

We know that a = mn, b = np + mq and c =pq

a) Ifa> 0 (positive) and c > 0 (positive),

i) when b > 0 (positive), thenp and q are positive,

ii) when b < 0 (negative), thenp and q are negative.

qnxpmxcbxax 2

pqxnpmqmnx

pqnpxmqxmnx

2

2

nx

mx p

pq

nx p

(np + mq)x

q

2mnx

mx q

Continue to next slide


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Factorizationb) Ifa> 0 (positive) and b and c have opposite signs, thenp and

q are also of opposite signs because m and n are positive.

If , then

i) ifb is positive, q is positive,ii) ifb is negative, q is negative.

If , then

i) ifb is positive, q is negative,

ii) ifb is negative, q is positive.

Continue to next slide

pqxnpmqmnxcbxax 22

npmq

mqnp


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Example 6

Solve the quadratic equations below by using factorization

method.

a) b)

c)

Solution:a)

3x + 2 = 0 or x2 =0

x = x = 2

0443 2 xx 0101962 xx

0443 2 xx

0223 xxx

3x +2

-4

+2x

-4x

-2

23x

-6x

3

2

0164 2 x


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Example 7b)

-3x + 2 = 0 or 2x5 =0

x = x =

010196

2

xx 05223 xx

3

2

2x

-3x +2

-10

+4x

19x

-5

2

6x

+15x

2

5


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Example 8c)

2x+ 4 = 0 or 2x - 4 =0

x = x =

Alternate method:

0164

2

x 04242 xx

2

2x

2x +4

-10

+8x

0

-4

2

4x

-8x

2

0164 2 x

2

4

4

164

2

2

x

x

x

x


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NoteIfu is an algebraic expression and dis a positive real

number, u2 = dhas exactly two solutions

Ifu2 = d, then u = oru = -

Equivalently,

Ifu2 = d, then u =

E.g. 1:

5x2

= 20x2 =

x =

x =2

d

E.g. 2:

(x - 2)

2

6 = 0(x - 2)2 = 6

x2 =

x = 2

d

d

45

20

6

6


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Converting a quadratic polynomial tothe form of .

Ifx2 + bx is a binomial, then by adding , which is the

square of half the coefficient ofx, a perfect square

trinomial will result.That is,

Note:

If , then

Completing the Square

cbxaxxp 2

)(qpxaxp 2)()( 2

2

b

bxx222

22

bx

b

du 2 du


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Example 9 Solve by using completing the square.

Solution:

0362 xx

362 xx

0362 xx

222 3336 xx

123 2 x

123 x

323x

22

2

2

632

66

xx


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Example 10 Solve by using completing the square.

Solution:

0323 2 xx

03

3

3

23

2

xx

0323 2 xx

01

3

22 xx

13

22 xx

22

2

23

2

123

2

32

xx

22

2

311

31

32

xx

Continue at next slide


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Example 10 (cont.)

9

10

3

12

x

9

10

3

1x

3

101

3

10

3

1

9

10

3

1

x

x

x


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Quadratic formula

For any quadratic equations , the roots aregiven by

02

cbxax

a

acbbx

2

42


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Example 11 Solve by using quadratic formula.

Solution:

From the equation, we know,

a = 1, b = -7 and c = 12

Then,

01272 xx

12

1214772

x

2

48497 x

2

17x

4

2

17

x

x or

3

2

17

x

x


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Example 12 Solve by using quadratic formula.

Solution:

From the equation, we know,

a = 4, b = 12 and c = 7

Then,

07124 2 xx

42

74412122

x

8

11214412 x

8

3212 x

2

23

8

2412

x

x or

8

21612 x

8

21612 x

8

2412 x

2

23

8

2412

x

x


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Exercise 61. Solve the following equations

a. b.

2. Solve the equations by using the factorization method.

a)

b)c)

d)

3. Solve the equations by completing the square, giving the

solutions in surd form.a)

b)

c)

022 xx

0253

2

xx 092 xx9412 2 xx

0635 2 xx0652 xx

012 2 xx

092 2 x 0473 2 x


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3. Solve the equations by using quadratic formula.

a)

b)

c)

d)

Exercise 6 (cont.)

0169 2 xx

0842 xx

0962 2 xx

0425 2 xx

ee004 fhs lnt 003 chapter 1 jan11

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