Download Chapter #6

Chapter #6
Exponents and Polynomials
In order for us to be successful in math we must get very comfortable when using variables in
expressions and in our equations. One of the first steps in becoming comfortable with variables is to
practice writing exponents with variables and then using our skills with exponents to manipulate
polynomial operations. Chapter 6 starts with learning the rules of exponents and ends with us learning
operations on polynomials. Take your time with this section and make sure you do lots of practice
problems. The better you get with using exponents and polynomials the easier math will become.
Section 6.1 Simplifying Integers Exponents I
Objectives
•
•
Simplify expressions by using properties of integer exponents.
Recognize which property of exponents is used to simplify an expression.
Instruct:
1. What do we mean when we say exponent?
2. Given 53 the 5 is called the ___________ and the 3 is called the __________.
3. Does 50 = 0 TRUE/FALSE
4. The expression 00 is _____________
5. Section 6.1 discusses five important properties of exponents. Please list then by name and then
explain in your own words how each property works.
1) _____________________________
1
2) _____________________________
3) ______________________________
4) _____________________________
5) ______________________________
Practice
1. Simplify the following
a.
c.
x 0 + (−2 y )0 =
b.
−12 x −3 y
=
4 x −2 y 4
( −5a b )( 6a c ) =
2 5
3
Section 6.2a: Simplifying Integer Exponents II
Objectives
•
Simplify powers of expressions by using the properties of integer exponents
Instruct:
1. If “a” is a nonzero number and the variables m and n are integers, then
m n
(a )
= a mn
represents which rule of exponents?
2
2. By using the Power of a Product rule we get
( 5a )
2
= 5a 2 . TRUE/FALSE.
3. What are the two basic shortcuts that can be used with negative exponents and fractions?
Practice
1. Simplify the following expression and write your answer with positive exponents:
 ( 3 x −1 ) 2   2 
  5x 
a. 
3
 25 y   6 y −2 


 a 
b. 

 −3b 
−2
Section 6.2b: Scientific Notation
Objectives
•
•
•
Write decimal numbers in scientific notation.
Write numbers in scientific notation as decimal numbers.
Perform operations with decimal numbers by using scientific notation
Instruct:
1. Scientific Notation is used to write very large or very small numbers as the product of a number
greater than or equal to 1 and less than 10 and an integer power of 10. TRUE/FALSE
2. Is 0.000000069 = 69 × 10−9 written in correct scientific notation? Why or why not?
3. Since multiplying 106 is the same as multiplying by 1,000,000, what is multiplying by 10−6 the
same as?
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Practice
1. Write 2.758 × 105 in decimal form
2. Write 0.09651 in scientific notation
Section 6.3a: Identifying Polynomials
Objectives
•
•
•
Define a polynomial.
Classify a polynomial as a monomial, binomial, trinomial, or a polynomial with more than three
terms.
Identify the degree and leading coefficient of a polynomial.
Instruct:
1. Write the definition of a term.
2. What is a constant term?
3. The number in front of a variable in a term is called the _________________ of the variable.
4. We classify polynomials by how many terms each expression has. Monomial have ____ term,
Binomial have ____ terms, Trinomial have ___ terms and any expression with more than
three terms we will just say it is a polynomial.
4
Practice
1. Simplify each of the following polynomials by combining like terms. Then write the polynomials
in descending order and tell the degree and type of the polynomial.
a.
3x4 − 7 x4 =
b.
4 y 2 + 5 y − 10 − y 2 =
Section 6.3b: Evaluating Polynomials
Objectives
•
Evaluate a polynomial for given values of the variable.
Instruct:
1. To evaluate a polynomial we need to be given a value (a number) for the variable and then we
rewrite the expression by replacing (substituting) the variable with that value where ever it occurs and
then we follow the rules of order of operations. True/False
Practice
1. Evaluate the monomial :
(− y)
3
when y = 4.
2. Evaluate the following trinomial at x = 5.
−10 x − 4 x 2 − 2
Section 6.4: Adding and Subtracting Polynomials
Objectives
•
•
•
Add polynomials
Subtract polynomials
Simplify expressions by removing grouping symbols and combining like terms.
5
Instruct:
1. The sum of two or more polynomials is found by _______________________________.
2. A negative sign written in front of a polynomial in parentheses means take the opposite of the
entire polynomial by changing the sign of every term inside the parentheses. True/False
3. When adding or subtracting polynomials in a vertical format it is recommended to first rewrite
the polynomial in ______________ order.
Practice
1. Perform the indicated operations by removing the parentheses and combining like terms.
a.
(7 x − y ) + ( x + 3 − 2 y ) =
b.
( −a
c.
( −5 x + 6 ) − (6 − 5 x) =
2
+ 9 + a ) − ( − a + 9) =
6
Section 6.5a Multiplying a Polynomial by a Monomial
Objectives
•
Multiply a polynomial by a monomial.
Instruct:
−4 x (9 x 2 − 5) = −36 x 3 + 20 x is an example of how we use the __________________property
1.
to multiply each term of a _______________ by the________________.
2. Explain in your own words the Commutative Property of Multiplication
Practice
1. Consider the following multiplication problems. Then use the distributive property and /or the
product rule of exponents to multiply the polynomials.
a.
( −2 x
b.
( 7 x )( 2 x
2
3
+ 5 ) ( −4 x ) =
2
− 2 x + 3) =
Section 6.5b Multiplying Two Polynomials
Objectives
•
Multiply polynomials.
Instruct:
1. When using the distributive property to multiply two binomials what is a quick way to check if
your products are correct?
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2. The distributive property is applied by___________________ of one polynomial by
____________ of the other.
Practice
1. First multiply the following polynomials using the distributive property and then simplify by
combining like terms:
a.
c.
( x − 7 )( x + 7 ) =
b. ( x + 7 )( x + 7 ) =
( x − 2 ) ( 3x 2 − x + 4 ) =
Section 6.6a The FOIL Method
Objectives
•
Multiply two binomials using the FOIL method
Instruct:
As you work through the problems pay close attention to how the FOIL method works. We will be
using this method later in reverse when we factor polynomials. So it is important that you
understand how this works.
1. What do each of the letters represent in FOIL when we multiply two binomials using the FOIL
Method?
F
O
I
L
8
Practice
1. Multiply the following binomials by using the FOIL method. Make sure you simplify your
answer by combining like terms.
a.
( x + 3)( 4 x − 2 ) =
2. A rectangle has a length of 8 inches and width of 5 inches. A small square, x inches on each
side, is cut out from each corner of the original rectangle.
a. Draw a picture of the rectangle and label its dimensions, please include the cut out
corner in your drawing.
b. Find the Area of the remaining portion of the rectangle after we remove the small
squares in the corners.
Section 6.6b Special Products
Objectives
•
•
Multiply binomials, finding products that are the difference of squares.
Square binomials, finding products that are perfect trinomials.
9
Instruct:
1. When multiplying two binomials in the form of sum and differences of the same two terms,
2
such as: ( x − 2 ) ( x + 2) = x − 4
we use
The __________________ of Two Squares
2. When multiplying two binomials as in #1 above, the product will only have two terms
because the second and third terms from FOIL cancel out. True/False
3. When we multiply two binomials that are the same we call the product a _________________
Trinomial.
4. The following two expressions represent two different forms of a Perfect Square Trinomial.
What are they called
2
= x 2 − 2ax + a 2 __________________________________________
2
= x 2 + 2ax + a 2 ___________________________________________
a.
( x − a)
b.
( x + a)
Practice
1. Simplify the following.
a.
c.
( x + 2)
( y − 3)
2
2
=
b.
( b − 4 )( b + 4 ) =
=
10
Section 6.7a: Division by a Monomial
Objectives
•
Divide polynomials by monomials
Instruct:
1. What is a rational expression ? (the definition will work here)
2. To divide a polynomial by a monomial we simply separate the terms and divide each term in the
_________________ by the monomial in the __________________.
Practice
1. Simplify the expressions by first dividing the following polynomials
a.
10 x 2 − 5 x
=
5x
b.
=
11
Chapter #7
Factoring Polynomials and Solving Quadratic Equations
In the previous chapter we combined polynomials with addition, subtraction, and multiplication. This
chapter concentrates on pulling apart polynomials through factoring. There are four basic factoring
techniques explored in this chapter; factoring greatest common factors, factoring by grouping, factoring
by trial and error, and factoring with special formulas. One common skill used to solve equations,
especially quadratic equations, is to factor. This skill is explained at the end of the chapter.
Section 7.1a: Greatest Common Factor of Two or More Terms
Objective
•
Find the greatest common factor of a set of terms.
Instruct:
1.
Give an example of a product and its factors.
2.
Listing the common factors and then choosing the greatest of these can be inefficient and time
consuming. A better way is to use ___________ ________________ . (write in the twoword title for this method)
3.
Write the three words that are abbreviated by the expression GCF.
4.
If, when finding the GCF for any set of integers or algebraic terms with integer coefficients, you
discover there is no common prime factor or variable, then what is the GCF?
12
Practice
1. Find the GCF for each of the sets of algebraic terms.
a) 48, 24, 64
b) 6mn, 8m3 n, 2mn2 p
Section 7.1b: Greatest Common Factor of a Polynomial
Objective
•
Factor polynomials by finding the greatest common monomial factor.
Instruct:
1. 5k + 15 = 5(k + 3) is an example of what property?
2. True or False.
-10a4 b + 15a4 = 5a4 (-2b + 3) and -10a4 b + 15a4 = -5a4 (2b - 3) are both correct
factorizations.
13
Practice
1. Factor the given polynomial by finding the greatest common monomial factor (or the negative
of the greatest common monomial factor)
a)
45 + 15x2
b)
-3m2n + 6mn – 3mn2
2. Simplify the expression. Assume no denominator is equal to 0.
12k 6 m
−3k 2 m
Section 7.1c: Factoring Expressions by Grouping
Objective
•
Factor polynomials by grouping.
Instruct:
1. In the previous two sections we factored out monomials. Now, we are factoring out binomials
or other polynomials. Give an example of a binomial.
2. Consider the polynomial, 6x2(4x + 3) – 2(4x + 3). Write down the binomial that is common to
both terms in the polynomial.
3. How many terms should a polynomial contain if you are going to factor by grouping?
14
Practice
1. Write down the binomial that is common in both terms of the following expressions
a)
3x(a + b) + 4(a + b)
b)
4k3(y – 4) – m(y – 4)
2. Factor each expression by factoring out the common binomial.
k(b2 + 1) – (b2 + 1)
3. Completely factor the expression by grouping.
3x – 12 + xy – 4y
Section 7.2: Factoring Trinomials: Leading Coefficient 1
Objective
•
•
Factor trinomials with a leading coefficient of 1 (of the form x2 + bx + c).
Factor out a common monomial factor and then factor such trinomials.
15
Instruct:
1. In the previous chapter you learned to multiply two binomials by using the FOIL method. Write
down the word associated with each letter in the word FOIL.
F = __________________
O = __________________
I = ___________________
L = ___________________
2. To factor a trinomial with leading coefficient of 1, find two factors of the ____________ term
whose sum is the coefficient of the ___________ term.
3. Finding the two factors to use when factoring a trinomial with leading coefficient “1”, is
considered by many to be a trial and error method. Find two factors whose product is -20
and whose sum is -8.
4. Considering the signs of the constant (last) term of the trinomial, x2 + bx + c, along with the sign
of the coefficient of the middle term, can help speed up the trial and error method.
If the sign of the constant term, c, is _____________ then both factors must have the same
sign.
a. Both are ___________ if b is positive.
b. Both are ___________ if b is negative.
If c is ________________ , then one factor must be negative and one factor must be
positive.
5. Always remember, when factoring a polynomial, to look for a common _____________ factor
first. If there is one, be sure to include it with your final answer.
16
Practice
1. For the two numbers listed, find two factors of the first number such that their product is the
first number and their sum is the second number. List these two factors.
a)
30, 17
b)
54, -15
2. Factor each trinomial given by the trial and error method.
a)
k2 + 3k – 15
b)
a2 + 5a + 4
3. Factor the trinomial by the trial and error method.
3x2 – 36x + 105
17
Section 7.3a: Factoring Trinomials by Grouping
Objective
•
Factor trinomials using the ac-method.
Instruct:
1. In the previous section you learned about factoring a trinomial, ax2 + bx + c, by the trial and
error method. This method is the reverse of multiplying binomials by FOIL. This section teaches
you the ac-method, which is factoring by _________________.
2. List the steps to factor a trinomial, ax2 + bx + c, by the ac-method.
a) Multiply ___________.
b) Find two integers whose product is _________ and whose sum is __________.
c) Rewrite what term, ax2, bx, or c, by using the numbers found in part (b).
d) Factor by ______________ the first two terms and the last two terms.
e) Finish by factoring out the common ____________ factor to find two binomial factors for
your trinomial.
Practice
1. Completely factor the following trinomials.
a)
4m2 - 11m – 3
b)
9w3 – 2w2 – 7w
18
Section 7.3b: Factoring Trinomials by Trial and Error
Objective
•
Factor trinomials by using reverse FOIL (or trial-and-error method).
This is a good lesson to practice some factoring techniques. This section has not been assigned, so you
do not need to complete 7.3b.
Section 7.4: Special Factorizations - Squares
Objective
•
•
•
Factor the difference of two squares.
Factor perfect square trinomials.
Complete the square of trinomials by determining the missing terms that make incomplete
trinomials perfect square.
Instruct:
1. In the previous chapter you learned about three special products of binomials. Write down the
names of these three special products. Note that two of them have the same name.
Special Product
(a + b)(a – b) = a2 – b2
Name
_________________________
(a + b)2 = a2 + 2ab + b2
_________________________
(a – b)2 = a2 – 2ab + b2
_________________________
2. Give an example of a perfect square trinomial.
3. Give an example of a polynomial that is the difference of two squares.
19
4. Complete the following summary steps from slide 9 of Instruct.
1. Look for a common factor.
2. Check the number of terms.
i. Two terms
ii. Three terms
iii. Four terms
3. Check the possibility of factoring any of the factors.
5. When you are completing the square of a trinomial and trying to find the last (constant) term,
take ____________ of the middle term and ______________ it.
When finding the middle term, take the square root of the last (constant) term and
____________it by 2. Remember of course to include the variable.
Practice
1. Indicate (yes or no) if the given polynomial has a special factorization. If it does, state whether it
is a perfect square trinomial or the difference of two squares
a)
9m2 – 16k2
b)
a2 – 6a + 9
20
c)
q2 + 4p2
2. Complete the square by adding the correct missing term on the left, then factor as indicated.
X2 + _________ + 36 = (______________)2
3. Factor the polynomial using special factorization.
25x2 – 36y2
Section 7.5: Solving Quadratic Equations by Factoring
Objective
•
Solve quadratic equations by factoring.
Instruct:
1. Write down the Zero- factor property.
For any real numbers a and b, if ________________________________________________.
2. Finish the following sentences.
To use the factorization principle to solve a quadratic equation:
1.
Factor __________________________________________________________.
21
2. Set _____________________________________________________________.
3. Is the first step when solving this quadratic equation to factor an x from each term on the left
hand side, or to add 8 to each side of the equation?
3x2 – 10x = -8
Practice
1. Solve the following equations by factoring.
a)
4m2 - 11m – 3 = 0
c)
9x2 – 1 = 0
b)
9w2 = 2w + 7
Section 7.6: Applications of Quadratic Equations
Objective
•
Solve word problems by writing quadratic equations that can be factored and solved.
This is a good lesson to see how we can use factoring to solve problems. This section has not been
assigned, so you do not need to complete 7.6. This kind of material will be covered in math 95.
Chapter #8
Rational Expressions
22
This chapter concentrates on working with rational expressions. You have been adding, subtracting,
multiplying, dividing, and factoring polynomials previous two chapters. Now we concentrate our
attention on the quotient of two polynomials. After performing some arithmetic on these expressions,
we solve some equations involving them and finish with some applications. Working with factions can
be challenging for some folks, so work hard and stay persistent throughout the chapter.
Section 8.1a: Defining Rational Expressions
Objective
Find the numerical value of a rational expression.
Determine what values of the variable, if any, will make a rational expression undefined.
•
•
Instruct:
1. Give an example of a rational expression.
2. Describe the process you would use in order to determine the values of the variable that make a
rational expression undefined.
3. Using the example you wrote down in #1, determine the values of the variable that make the
rational expression undefined. Be sure you are using the process you outlined in #2.
4. If only the numerator of a fraction is equal to zero, the fraction is ________________?
If only the denominator of a fraction is equal to zero, the fraction is ________________?
Practice
1. Find the restricted value(s) of x for the given rational expression.
a)
x +3
x −5
b)
4x − 5
x 2 − 16
23
2. Consider the following rational expression, evaluate the rational expression when x = 2.
3x + 1
2x2 + x − 5
Section 8.1b: Reducing Rational Expressions
Objective
•
Reduce rational expression to lowest terms.
Instruct:
1.
The ____________________________________ is used to build a rational expression to
higher terms or to reduce a rational expression.
2. A rational expression is reduced to lowest terms if the greatest common factor (GCF) of the
numerator and denominator is equal to _______________.
3. When reducing a rational expression to lowest terms, can you “divide out” common factors or
terms?
4. List the three locations a negative sign can be written on a fraction without changing the value
of the fractions.
24
Practice
1. Reduce the following rational expressions to its lowest terms.
3x
a)
9 x − 15
b)
x 2 − x − 20
x 2 − 16
Section 8.2: Multiplication and Division of Rational Expressions
Objective
•
•
Multiply rational expressions.
Divide rational expressions.
This is a good lesson to see how we can use factoring to solve problems. This section has not been
assigned, so you do not need to complete 8.2.
Section 8.3: Addition and Subtraction with Rational Expressions
Objective
•
•
Add rational expressions.
Subtract rational expressions.
This lesson explains how rational expressions can be added or subtracted. As with numerical fractions
this is done by finding a common denominator. Feel free to peruse this lesson, but it has not been
assigned.
25
Section 8.4: Complex Algebraic Fractions
Objective
•
Simplify complex fractions and complex algebraic expressions.
This lesson explains how to deal with complex algebraic fractions and expressions. These are the
fractions that have fractions in either the numerator or denominator or both. That is, these are
fractions with fractions. Again, feel free to check it out, but this lesson has not been assigned.
Section 8.5: Solving Equations Involving Rational Expressions
Objective
•
•
•
Solve equations involving rational expressions.
Solve proportions.
Solve word problems by using proportions and rational expressions.
Instruct:
1.
One approach to solving an equation containing fractions is the “clear” the fractions. This is
done by multiplying all terms on both sides of the equation by the ___________ of the
denominators.
2.
Write down the four steps to solve and equation containing rational expressions.
a. Find ….
b.
26
a.
b.
2. Write down the definition of a proportion and give an example of one.
3. Care must be taken with the units in the setup of a proportion. Write down the two conditions,
regarding the units, for which one must be true. (slide #20)
a.
b.
Practice
1. Considering the following equation, state the restrictions on the variable. Don’t solve the
equation.
a−5
2
+3=
a
a−3
2. Solve the following rational equation.
−3
2
+
=1
x −3 x +7
3. Doug is able to drive 420 miles on 20 gallons of gasoline in his Volkswagen Microbus. If he can
maintain this fuel efficiency, how many gallons must he use to drive 1,155 miles?
27
Section 8.6: Applications Involving Rational Expressions
Objectives
•
•
•
Solve applications related to fractions.
Solve applied problems related to work.
Solve applications involving distance, rate, and time.
This lesson deals with applications of rational expressions (fractions). There are several good
applications you might want to investigate. However, this lesson has not been assigned.
Section 8.7: Additional Applications: Variation
Objective
•
•
Understand direct, inverse, and combined variation.
Know how to solve applied problems, dealing with direct variation, inverse variation, and
combined variation.
This lesson deals with specific applications of rational expressions (fractions) concerning direct and
indirect variation. You might want to investigate. This lesson has not been assigned, however, it is a
bonus lesson and you can earn bonus points for completing it.
Chapter #9
Real Numbers and Radicals
The real number system is not complete without studying the powerful math tool we call a Radical.
Remember math tools often come in pairs. Multiplication and division work as a team to undo order of
operation. Adding and subtracting also will undo each other. Some of you have previously learned
that exponents and radicals work together in the same similar pattern. Some of you will find this
concept new. Chapter 9 is all about how we can manipulate radicals in our expressions, equations and
word problems. This is a interesting subject. Pay attention to details. We use radicals in many different
forms in math and it is important you get comfortable with them.
Section 9.1 Evaluating Radicals
Objectives
•
•
•
Determine if a real number is rational or irrational
Understand the meaning of radical expressions
Simplify radicals
28
Instruct:
1. When we square an integer or a rational number the result is a perfect square.
a. Draw a picture of a perfect square with sides of 5. Then square the length of one side to
find the area. What number do you get?
b. Take you answer to part 1a and square root it. What number do you get? What do
you notice about your answer when comparing it to part 1a?
2. When we cube a number the cube root of the result is a perfect cube.
a. Draw a picture of a box with the length, width, and height all equal to 3. Then cube the
value of one of the sides to find the volume. What number do you get?
b. Take your answer to part 2a and cube root it. What number do you get?
3. Is it possible to find the square root of a negative number and get a real number answer? Yes or No.
If your answer is “no” , explain.
29
Practice
1. For each part below, evaluate the radical expression . If the result is irrational please round your
answers to two decimals places. If the answer is imaginary please write, “Not a Real number“
a.
120 =
b.
−9 =
c.
49 =
d.
3
−27 =
Section 9.2: Simplifying Radicals
Objectives
•
Simplify radicals, including square roots and cube roots
Instruct:
1. What are the two properties of square root and how do we use them to simplify a radical?
2. When are square roots and cube roots considered to be in simplest form?
3. When dealing with radicals with variables is the value of our variable always positive? Yes or
No
30
4. To avoid confusion we will assume the variables under the radical sign will represent only
___________________________.
5. When simplifying square roots with even powers of variables, we look for even exponents by
__________________ the exponent by _________________.
6. When simplifying square roots with odd powers of variables, we factor the expression into two
terms, one with _______________ and the other with an __________________ exponent.
7. There are two approaches to simplifying radical expressions in the form of a single fraction in
the denominator. What are these two methods?
a. Method 1: ______________________________________________________
b. Method 2: _______________________________________________________
Practice
1. Simplify the following expressions:
a.
−
b.
3
18
=
25
27 x 9 y 4 =
Section 9.3a: Addition and Subtraction of Radicals
Objectives
•
Addition and subtraction of Radicals
31
Instruct:
1. Like radicals are terms that have the same ______________ or can be __________________ so
that the _______________ are the same.
2. When adding and subtracting radicals we simplify the radicals first and then we combine like
terms. True or False
3. What is the three step method for adding and subtracting radicals?
a. Step 1___________________________________________________
b. Step 2___________________________________________________
c. Step 3 ___________________________________________________
Practice
1. Simplify the following radicals
a.
3
5 − 9 3 5 + 3 125 =
b. 7 2 + 8 − 3 y 3 =
Section 9.3b: Multiplication of Radicals
Objectives
•
Multiply radical expressions.
Instruct:
1. We multiply radicals by multiplying the numbers or variables under the root and then
simplifying our radical answers. True or False
32
2. We multiply a more complex radical expression such as
(
2 5− 2
)
by using the
__________________ property.
3. To find the product of two binomials expression that contain radicals we multiply the
expressions like they were ordinary polynomials by using the ______________ method.
Practice
1. Multiply and simplify the following radical expression. Do not round your answers. All answers
need to be exact. Radical symbols in your answers are acceptable.
( 3 )( 6 ) =
a.
c.
(
3+4
)(
(
b.
)
2 5− 2 =
)
3+4 =
Section 9.4a Rationalizing Denominators
Objective
• After completing this section you will be able to rationalize the denominators of rational
expressions containing radicals.
Instruct:
1. A useful Algebra technique is to multiply by 1. In this section the form of 1 will take on one of three
forms:
1=
b
b
or
1=
a ±
b
a ±
b
or
1=
a ±
b
a ±
b
2. What would you multiply by to rationalize the denominators in the following examples?
a)
5
6
b)
3 2
5
c)
5
2− 3
Practice:
33
1. Rationalize the denominators of the rational expressions in #2 above.
Section 9.4b Division of Radicals
Objective
• Use the quotient rule for radicals along with rationalizing the denominator to simply fractions
containing radicals.
Instruct:
n
1. The quotient rule for radicals states that:
n
a
=
b
n
a
b
Practice:
1. Simplify the following expressions. Be sure that your answer has a rational denominator. (That is,
make there is not a radical in the denominator)
a)
14
28
b)
30
6
c)
84
44
Section 9.5 Solving Equations with Radicals
Objective
• Solve equations with radical expressions by squaring both sides of the equation.
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Instruct:
1. When solving radical equations, we will square both sides of the equation. This makes it extremely
important to check all answers with the original equation. Why is that?
2. Occasionally we will get an “extra” equation. What is this solution called? ______________________
Practice:
1. Solve the following radical equation; be sure to check for extraneous roots.
x+1=
x+7
2. Solve the following radical equation; be sure to check for extraneous roots.
5 + 2x - 3 = 8
Section 9.6 Rational Exponents
Objective
• Write radical expressions with fractional exponents in radical form.
• Simply expressions with fractional exponents.
• Evaluate roots or fractional exponents with the calculator.
This lesson deals with specific applications of rational exponents. That is, exponents that are fractions
what could that mean? You might want to investigate. This lesson has not been assigned,
however, it is a bonus lesson and you can earn bonus points for completing it.
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Section 9.7 The Pythagorean Theorem
Objective
• Find the distance between two points.
• Determine if triangles are right triangles given the coordinates of the vertices using the distance
formula and the Pythagorean Theorem.
• Show that specific geometric properties are true by using the distance formula.
• Find the perimeter of triangles given the coordinates of the vertices using the distance formula.
This lesson is mainly concerned with a useful formula called the "Distance Formula" that is useful in
determining the distance between two points. The Pythagorean Theorem is introduced in this lesson,
but is discussed more in chapter 10. Again, feel free to investigate this lesson, but it is not assigned
and will be covered in more detail in math 9
Chapter #10
Quadratic Equations
A large part of this course has been the study of quadratic equations. We spent a great deal of time
learning to factor quadratic expressions and then equations so that we could use the factoring
technique to solve quadratic equations. However, many times it is not practical to factor quadratics as
the system does not always work in a straightforward manner. In this chapter we will other techniques
to solving quadratic equations.
Section 10.1 Quadratic Equations: The Square Root Method
Objectives
• Solve quadratic equations using the definition of square root.
• Solve applications related to right triangles and the Pythagorean Theorem.
Instruct:
1. When solving a quadratic equation by factoring we must first set the equation equal to __________ .
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2. If x2 = c, then x = ___
c
3. In a right triangle with legs “a” and “b” and hypotenuse “c”, the Pythagorean Theorem states:
a2 + b2 = _____
4. In a right triangle, the hypotenuse is ALWAYS the longest side. True or False
Practice:
1. NOTE: When solving a quadratic equation by the square root method be sure to isolate the squared
term before taking the square root.
2. Solve the following equations.
a) (x – 5)2 = 10
b) 4(x + 2)2 = 15
c) (2x – 5)2 = 25
Section 10.2 Quadratic Equations: Completing the Square
Objectives
• Determine the constant terms that will make incomplete trinomials perfect square trinomials.
• Solve quadratic equations by completing the square.
This is a good place to learn this useful “Algebra Procedure” that is occasionally used in future math
classes.
Section 10.3 Quadratic Equations: The Quadratic Formula
Objectives
• Write quadratic equations in standard form.
•
Identify the coefficients of quadratic equations in standard form.
•
Solve quadratic equations by using the quadratic formula.
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•
Determine the nature of the solutions (one Real, two Real, or no Real solutions) by using the
discriminant.
Instruct:
1. The development of the quadratic equation is shown on pages 3 and 4 of the instruct portion. It will
be hard to follow if you have not spent time in section 3.2. However, you should commit this formula to
memory. Write the formula below.
If ax2 + bx + c = 0, then x =
2. The instruct section mentions the “discriminant”. What expression is the discriminant?
3. What happens if the discriminant is negative?
Practice:
1. Use the quadratic formula to solve: 2x2 – 5x = 4. (Be sure to set this to zero first)
2. Find the discriminant for the following: 4x2 – 12x + 9 = 0
3. Based on the value of the discriminant in problem #2, how many Real solutions does the equation in
#2 have?
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