6.1 Using Properties of Exponents What you should learn: Goal1 Goal2 Use properties of exponents to...

6.16.1 Using Properties of Exponents

What you should learn:GoalGoal 11

GoalGoal 22

Use properties of exponents to evaluate and simplify expressions involving powers.

Use exponents and scientific notation to solve real-life problems.

6.1 Using Properties of Exponents6.1 Using Properties of Exponents

Product of Powers Property

nmnm xxx

ex) 97 xx 97x 16x

ex) 4yy 41y 5y

The Power of a Power Property

mnnm xx

ex) 46x 46x 24x

ex) 342 342 122

ex) 573 573 353

Power of a Product

nnn yxxy

ex) 35x 335 x 3125x

ex) 542y 5452 y

2032y

Write each expression with positive exponents only.

ex) 3x 3

1

x

Negative Exponents in Numerators and Denominators

nn

bb

1 and n

nb

b

1

ex) 2424

1

16

1

Use the Zero-Exponent Rule

ex) 03 1

The Zero-Exponent Property

10 b

ex) 0x 1

Divide by using the Quotient Rule

ex)

10

30

x

x 1030x

20x

The Quotient of Powers Property

nmn

m

bb

b

Simplify by using the Quotient of Powers Rule

ex)

3

2

x

3

3

2

x

8

3x

The Power of Quotient Property

n

nn

b

a

b

a

ex)

2

3

2

x

9

2 22 x

Simplify.

ex)

3

12

15

45

x

x 93x

ex)

yx

yx2

5

4

12

033 yx 33x

Simplify.

ex)

2

64 23 xx

ex)

2

35

4

412

x

xx 2

3

2

5

4

4

4

12

x

x

x

x

2

6

2

4 23 xx 23 32 xx

xx 33

Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section

Give an example of a quadratic equation in vertex form. What is the vertex of the graph of this

equation?

assignmentassignment

6.26.2 Evaluating and Graphing Polynomial Functions


GoalGoal 22

Evaluate a polynomial function

Graph a polynomial function.

6.2 Evaluating and Graphing Polynomial Functions6.2 Evaluating and Graphing Polynomial Functions

Polynomial- is a single term or sum of two or more terms containing variables in the numerator with whole number exponents.

6 x4 35xor or

6x 24 2 xxor

Polynomial- is a single term or sum of two or more terms containing variables in the numerator with whole number exponents.

It is customary to write the terms in the order of descending powers of the variables.

This is Standard Form of a polynomial.

6547 23 xxx

Monomials-polynomials with one term.

Example) 6 or 2x or34x

Binomials-polynomials with two terms

Example) 53 x

Trinomials-polynomials with three terms.

Example) 654 2 xx

The Degree of

If a does not equal zero,

then the degree of is n.

The degree of a nonzero constant is 0.

The constant “ 0 “ has nono defined degree.

nax

nax

Polynomial

Degree of the polynomial is the largest degree of its terms.Degree of the polynomial is the largest degree of its terms.

Example) 2x , has a degree of 1

Example) , has a degree of 2

, has a degree of 3

Degree of the number is the exponent of the variable..

24x

nax

xxx 754 23 Example)

Classifying polynomials by degree

37x

654 2 xx

675 24 xxx

53 x

5 Constant,Constant,

Linear,Linear,

Quadratic,Quadratic,

Degree 0,

Degree 1,

Degree 2,

Degree 3,

Degree 4,

Monomial

Binomial

Trinomial

Monomial

Polynomial

Cubic,Cubic,

Quartic,Quartic,

Directions: Use Direct Substitution to evaluate the Polynomial Function for the given value of x.

7582 24 xxxf (x) = , when x = 3

7)3(5)3(8)3(2 24 f (3) =

Make the Substitution.

71572162

98

GoalGoal 11 Evaluate a polynomial function

Synthetic Substitution

1. Arrange polynomials in descending powers, with a 0 coefficient for any missing term.

)7(5)8(02 234 xxxx

Directions: Use Synthetic Substitution to evaluate the Polynomial Function for the given value of x.

Another way to evaluate a polynomial function is to use Synthetic Substitution.

NOTICE


)7(5)8(02 234 xxxx


)7(5)8(02 234 xxxx

3x-value

Polynomial in standard form

2 0 -8 5 -7

2 6 10 35 98

6 18 30 105

multiply

add


Which term of a polynomial function is most important in determining the end behavior of the

function?


6.36.3 Adding, Subtracting , and Multiplying Polynomials

What you should learn:GoalGoal 11 Add, subtract, and multiply

polynomials

6.3 Adding, Subtracting, and Multiplying6.3 Adding, Subtracting, and Multiplying

Add or subtract as indicated

ex) )74()2( 22 xyyxxyyx

xyyx 82 2

ex) )436()57( 3434 yxyxyxyx 34 945 yxyx

Add or subtract as indicated

ex) )4678()2( 322333 yxyyxxyx

3223 56710 yxyyxx

ex) )536()( 322333 yxyyxxyx

3223 2535 yxyyxx

Add or subtract as indicated (vertically)

ex) 222 xyyx

22 84 xyyx

ex) 34 853 yxyx 34 945 yxyx (+)

22 76 xyyx 34 92 yxyx

Add or subtract as indicated (vertically)

ex) yxxy 22 23

yxxy 22 35

ex) 34 853 yxyx 34 945 yxyx (-)

yxxy 22 52 34 178 yxyx

(-)

ex) 972 xx

4x

Use a vertical format to find each product

36284 2 xx

xxx 97 23 +

36193 23 xxx

Multiplying Monomials

To multiply monomials, multiply the coefficients and then multiply the variables. Use the product rule for exponents to multiply the variables: Keep the variable and add the exponents.

ex) )4(2 2xx multiply the coefficients and multiply the variables

))(42( 2xx

38x

ex) )4)(3( 36 xx 912x

ex) )7

3)(

5

4( 27 xx 9

35

12x

ex) ))(2)(6( 64 xxx 1112x

Finding the product of the monomial and the polynomial

ex) )34(2 xx xx 68 2

ex) )24(3 22 yyy 34 612 yy

ex) )353(2 22 yyy 234 6106 yyy

Finding the product when neither is a monomial

ex) )34)(3( xx 91234 2 xxx

ex) )324)(2( 2 yyy

648324 223 yyyyy

9154 2 xx

664 23 yyy

Multiply by using the rule for finding the product of the sum and difference

ex) )5)(5( xx 25552 xxx

22 5x

The Product of the Sum and Difference of Two Terms

))(( BABA 22 BA

252 x

Multiply by using the rule for theSquare of a Binomial.

ex) 2)5( x 25552 xxx

25102 xx

The Product of the Sum of Two Terms2)( BA 22 2 BABA

Multiply by using the rule for theSquare of a Binomial.

ex) 2)5( x 25552 xxx

25102 xx

The Product of the Difference of Two Terms2)( BA 22 2 BABA

Using the FOIL Method to Multiply Binomials

ex) )34)(3( xx 91234 2 xxx

9154 2 xx

Find the ProductFind the Product

ex) )5)(4( 2 xyyx

2320 yx

ex) )( 33 babab

423 babba


ex) ))(( yxyx

22 yx

ex) 2)( ba

22 2 baba

))(( baba

Find the Product

ex) )12)(1( xyxy

122 22 xyxyyx

ex) 222 )( ba

4224 2 bbaa

))(( 2222 baba

12 22 xyyx


ex) )42)(3( 2 xxxy

126342 223 xxxyyxyx

ex) ))(( cbacba

222 2 cbcba

222 cbcacbcbabacaba


How do you add or subtract two polynomials?


6.46.4 Factoring and Solving Polynomial Equations


GoalGoal 22

Factor polynomial expressions

6.4 Factoring and Solving Polynomial Equations6.4 Factoring and Solving Polynomial Equations

Use Factoring to solve polynomial expressions

Factoring Monomials means finding two monomials whose product gives the original monomial.

Factoring is the process of writing a polynomial as the product of two or more polynomials.

ex) 230xCan be factored in a few different ways…

)6)(5( xxa.)

b.) )2)(15( xx)3)(10( 2xc.)

d.) )5)(6( xx

Find three factorizations for each monomial.Directions:

1.) 420x

2.)

3.)

615x

527x

)5)(4( 3xx )20)(( 22 xx

)10)(2( 22 xx

Find the greatest common factor.

1.) 36x

2.) 515x327x 33x

and 210x22x

GCF of 6 and 10

(or what # divides into 6 and 10 evenly)

GCF of 6 and 10

(or what # divides into 6 and 10 evenly)

When dealing with the variables, you take the variable with the smallest exponent as your GCF.When dealing with the variables, you take the variable with the smallest exponent as your GCF.

and

Factoring out the greatest common factor.

But, before we do that…do you remember the Distributive Property?

)32(5 xx

xx 1510 2

When factoring out the GCF, what we are going to do is UN-Distribute.

Factor each polynomial using the GCF.Factor each polynomial using the GCF.

xx 54 ex) )5( 3 xx

xx 217 2 ex) )3(7 xx

xxx 10515 23 ex)

)23(5 2 xxx

Factor each polynomial using the Greatest Common Binomial Factor.

Factor each polynomial using the Greatest Common Binomial Factor.

)5(3)5( xxxex)

)5)(3( xx

)23()23(7 xxxex)

)23)(17( xx

4)4(9 2 xxxex)

)4)(19( 2 xx

Factor by GroupingFactor by Grouping

1892 23 xxxEx 1)

)189()2( 23 xxx

)2(9)2(2 xxx

Factor-out GCF from each binomial

Factor-out GCF

)2)(9( 2 xx Factored by Grouping

Group into binomials

Factoring the Sum or Difference of 2 Cubes

1.)Factoring the Sum of Two Cubes:

2.) Factoring the Difference of 2 Cubes:

))(( 2233 BABABABA

))(( 2233 BABABABA

Example 1)83 x

))(( 2233 BABABABA

328

)42)(2( 2 xxx

)22)(2( 22 xxx

or

Sum

Example2)643 x

))(( 2233 BABABABA

3464

)164)(4( 2 xxx

)44)(4( 22 xxx

or

Example 3)273 x

))(( 2233 BABABABA

3327

)93)(3( 2 xxx

)33)(3( 22 xxx

or

Difference

Definition of a Quadratic Equation

A quadratic equation in x is an equation that can be written in the standard form

where a, b, and c are real numbers, with a = 0. A quadratic equation in x is also called a second-degree polynomial equation in x.

/

02 cbxax

The Zero-Product Principle

If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero.

If AB = 0, then A = 0 or B = 0.

example) 0)2)(5( xx

According to the principle, this product can be equal to zero if either

0)5( x 0)2( xor+5 +5

x = 5

+2 +2

x = 2

The resulting two statements indicate that the solutions are 5 and 2.

example) 0472 2 xxFactor the Trinomial using the methods we know.

0)12( x 0)4( xor

+1 +1

x = 1/2

- 4 - 4

x = - 4

The resulting two statements indicate that the solutions are 1/2 and - 4.

Solve a Quadratic Equation by Factoring

(2x )(x ) = 0- +1 4

2x = 1

example) 962 xxMove all terms to one side with zero on the other. Then factor.

0)3( x+3 +3

The resulting two statements indicate that the solutions are 3.

Solve a Quadratic Equation by Factoring

(x )(x ) = 0- -3 3

x = 3

0962 xx

The trinomial is a perfect square, so we only need to solve once.


How can you use the zero product property to solve polynomial equations of degree 3 or more?


6.56.5 The Remainder and Factor Theorems

What you should learn:GoalGoal 11 Divide polynomials and relate the

result to the remainder theorem and the factor theorem.

6.4 The Remainder and Factor Theorem6.4 The Remainder and Factor Theorem

Divide using the long division

ex)

3

21102

x

xx

21103 2 xxx

x

xx 32

x7 21

+ 7

217 x

0

Divide using the long division with Missing Terms

ex)

12

18 3

x

x100812 23 xxxx

23 48 xx 24x x0

xx 24 2

12 x

24x x2 1

12 x

0

Synthetic DivisionTo divide a polynomial by x - c

1. Arrange polynomials in descending powers, with a 0 coefficient for any missing term.

2. Write c for the divisor, x – c. To the right, write the coefficients of the dividend.

3 1 4 -5 5

)3()554( 23 xxxx

3. Write the leading coefficient of the dividend on the bottom row.

4. Multiply c (in this case, 3) times the value just written on the bottom row. Write the product in the next column in the 2nd row.

3 1 4 -5 5

1 4 -5 5 3

1

1

3

5. Add the values in the new column, writing the sum in the bottom row.

6. Repeat this series of multiplications and additions until all columns are filled in.

3 1 4 -5 5

1 4 -5 5 3

1

1

3

3

7

add

7

21 add

16

7. Use the numbers in the last row to write the quotient and remainder in fractional form.

The degree of the first term of the quotient is one less than the degree of the first term of the dividend.

The final value in this row is the remainder.

1 4 -5 5 3

1

3

7

add 21

16

48

53

5543 23 xxxx

3

531672

xxx


)1()24( 2 xxx

-1 1 4 -2

Example 1)

1

-1

3

-3

-5

1

53

xx


)2()75( 3 xxx

2 1 0 -5 7

Example 2)

1

2

2

4

-1

2

5122

xxx

-2

5

Factoring a Polynomial

918112)( 23 xxxxfExample 1)

given that f(-3) = 0.

-3 2 11 18 9-6 -15 -9

2 5 3 0multiply

Because f(-3) = 0, you know that (x -(-3)) or (x + 3) is a factor of f(x).

918112 23 xxx )352)(3( 2 xxx

Factoring a Polynomial

1892)( 23 xxxxfExample 2)

given that f(2) = 0.

2 1 -2 -9 182 0 -18

1 0 -9 0multiply

Because f(2) = 0, you know that (x -(2)) or (x - 2) is a factor of f(x).

1892 23 xxx )9)(2( 2 xx

)3)(3)(2( xxx


If f(x) is a polynomial that has x – a as a factor, what do you know about the value of f(a)?


6.66.6 Finding Rational Zeros

What you should learn:GoalGoal 11 Find the rational zeros of a

polynomial.

6.6 Finding Rational Zeros6.6 Finding Rational Zeros

Find the rational zeros of


The Rational Zero Theorem

0

0

at coefficien leading offactor

a ermconstant tfactor

q

p

12112)( 23 xxxxf

solution List the possible rational zeros. The leading coefficient is 1 and the constant term is -12. So, the possible rational zeros are:

1

12,

1

6,

1

4,

1

3,

1

2,

1

1x

Find the Rational Zeros of


30772)( 23 xxxxf

solution

List the possible rational zeros. The leading coefficient is 2 and the constant term is 30. So, the possible rational zeros are:

30,15,10,6,5,3,2,1,2

15,

2

5,

2

3,

2

1x

Example 1)

Notice that we don’t write the same numbers twice

-2 1 7 -4 -28

Example 1)

1

-2

5

-10

-14

)145)(2()( 2 xxxxf

28

0

Use Synthetic Division to decide which of the following are zeros of the function 1, -1, 2, -2

2847)( 23 xxxxf

)7)(2)(2()( xxxxfx = -2, 2

1 1 4 1 -6

Example 1)

1

1

5

5

6

)65)(1()( 2 xxxxf

6

0

Find all the REAL Zeros of the function.

64)( 23 xxxxf

)3)(2)(1()( xxxxfx = -2, -3, 1

2 1 1 1 -9 -10

Example 2)

1

2

3

6

7

14

5

Find all the Real Zeros of the function.

109)( 234 xxxxxf

10

0

-1 1 3 7 5

1

-1

2

-2

5

-5

0

)52)(1)(2()( 2 xxxxxf

x = 2, -1

-1 1 3 7 5

1

-1

2

-2

5

-5

0


How can you use the graph of a polynomial function to help determine its real roots?


6.76.7 Using the Fundamental Theorem of Algebra

What you should learn:GoalGoal 11 Use the fundamental theorem of

algebra to determine the number of zeros of a polynomial function.

6.7 Using the Fundamental Theorem of Algebra6.7 Using the Fundamental Theorem of Algebra

THE FUNDEMENTAL THEOREM OF ALGEBRA

If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers.

-5 1 5 -9 -45

Example 1)

1

-5

0

0

-9

)9)(5()( 2 xxxf

45

0

Find all the ZEROs of the polynomial function.

4595)( 23 xxxxf

)3)(3)(5()( xxxxfx = -5, -3, 3

Example 2)

)4)(3()( 22 xxxf

Find all the ZEROs of the polynomial function.

12)( 24 xxxf

3 1 0 1 0 -12

1

3

3

9

10

30

30

90

0

i2,3 )4)(3)(3()( 2 xxxxf

NOT DONE YET

Example 1)

Decide whether the given x-value is a zero of the function.

55)( 23 xxxxf , x = -5

5)5()5(5)5()5( 23 f

0)5( f

So, Yes the given x-valueis a zero of the function.

Example 1)

Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1.

)56)(4()( 2 xxxxf

-4, 1, 5

20192)( 23 xxxxf

)5)(1)(4()( xxxxf


How can you tell from the factored form of a polynomial function whether the function has a

repeated zero?


At least one of the factors will occur more than once.

6.86.8 Analyzing Graphs of Polynomial Functions

What you should learn:GoalGoal 11 Analyze the graph of a polynomial

function.

6.8 Analyzing Graphs of Polynomial Functions6.8 Analyzing Graphs of Polynomial Functions

Plot x-intercepts:

Find the Turning Points:

The y-coordinate of a turning point is a Local Maximum if the point is higher than all nearby points. The y-coordinate of a turning points is a Local Minimum if the point is lower that all nearby points.



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6.16.1 Using Properties of Exponents


GoalGoal 22

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hghjghjghjghjghjgjhb

6.1 Using Properties of Exponents6.1 Using Properties of Exponents



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