6.1 using properties of exponents what you should learn: goal1 goal2 use properties of exponents to...
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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
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
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
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
What you should learn:GoalGoal 11
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
Synthetic Substitution
)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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
Which term of a polynomial function is most important in determining the end behavior of the
function?
assignmentassignment
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
(-)
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
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
Find the ProductFind the Product
ex) )42)(3( 2 xxxy
126342 223 xxxyyxyx
ex) ))(( cbacba
222 2 cbcba
222 cbcacbcbabacaba
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
How do you add or subtract two polynomials?
assignmentassignment
6.46.4 Factoring and Solving Polynomial Equations
What you should learn:GoalGoal 11
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
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.
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
How can you use the zero product property to solve polynomial equations of degree 3 or more?
assignmentassignment
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 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
Synthetic DivisionTo divide a polynomial by x - c
)1()24( 2 xxx
-1 1 4 -2
Example 1)
1
-1
3
-3
-5
1
53
xx
Synthetic DivisionTo divide a polynomial by x - c
)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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
If f(x) is a polynomial that has x – a as a factor, what do you know about the value of f(a)?
assignmentassignment
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
6.6 Finding Rational Zeros6.6 Finding Rational Zeros
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
6.6 Finding Rational Zeros6.6 Finding Rational Zeros
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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
How can you use the graph of a polynomial function to help determine its real roots?
assignmentassignment
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
Reflection on the SectionReflection on the SectionReflection on the SectionReflection on the Section
How can you tell from the factored form of a polynomial function whether the function has a
repeated zero?
assignmentassignment
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.
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
6.16.1 Using Properties of Exponents
What you should learn:GoalGoal 11
GoalGoal 22
ghghhhghjghjghghggghjg
hghjghjghjghjghjgjhb
6.1 Using Properties of Exponents6.1 Using Properties of Exponents
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