STANDARDS OF LEARNING
CONTENT REVIEW NOTES
ALGEBRA I
rd
3 Nine Weeks, 2016-2017
1
OVERVIEW
Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a
resource for students and parents. Each nine weeks’ Standards of Learning (SOLs) have been identified and a
detailed explanation of the specific SOL is provided. Specific notes have also been included in this document
to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models
for solving various types of problems. A “
” section has also been developed to provide students with
the opportunity to solve similar problems and check their answers. The answers to the “
are found at the end of the document.
” problems
The document is a compilation of information found in the Virginia Department of Education (VDOE)
Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE
information, Prentice Hall textbook series and resources have been used. Finally, information from various
websites is included. The websites are listed with the information as it appears in the document.
Supplemental online information can be accessed by scanning QR codes throughout the document. These will
take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the
document to allow students to check their readiness for the nine-weeks test.
To access the database of online resources scan this QR code,
The Algebra I Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of
questions per reporting category, and the corresponding SOLs.
Algebra I Blueprint Summary Table
Reporting Categories
Expressions & Operations
No. of Items
12
Equations & Inequalities
18
Functions & Statistics
20
Total Number of Operational Items
Field-Test Items*
Total Number of Items
SOL
A.1
A.2a – c
A.3
A.4a – f
A.5a – d
A.6a – b
A.7a – f
A.8
A.9
A.10
A.11
50
10
60
* These field-test items will not be used to compute the students’ scores on the test.
It is the Mathematics Instructors’ desire that students and parents will use this document as a tool toward the
students’ success on the end-of-year assessment.
2
3
Laws of Exponents & Polynomial Operations
A.2 The student will perform operations on polynomials, including
a) applying the laws of exponents to perform operations on expressions;
Monomial is a single term. It could refer to a number, a variable, or a product of a
number and one or more variables.
Some examples of monomials include:
When you multiply monomials that have a common base, you add the exponents.
Example 1: Multiply
This works because when you raise a number or variable to a power, it is like
multiplying it by itself that many times. When you then multiply this by another power of
the same number or variable, you are just multiplying it by itself that many more times.
Example 2:
.
Scan this QR code to go
to a video tutorial on
multiplying monomials!
Example 3: Simplify
When you raise a power to a power, you multiply the exponents.
This means 3² times itself 4 times!
Example 4: Simplify
4
Example 5: Simplify
Often, you will be asked to multiply monomials and raise powers to a power. Make
sure that you follow the ORDER OF OPERATIONS! Raise to powers first, then
multiply.
Example 6: Simplify
Laws of Exponents
Simplify each expression
1.
2.
3.
4.
When you divide monomials with like bases, you will subtract the exponents.
Anything raised to the zero power is equal to ONE!
To find the power of a quotient, raise both the numerator and the denominator to the
power. (Remember to follow the order of operations!)
5
Scan this QR code to go
to a video tutorial on
dividing monomials!
Example 7:
Example 8:
Remember that anything to
the zero power equals 1!
You will also see negative exponents in monomials. When you have a negative
exponent, you will reciprocate that variable (move it to the other side of the fraction
bar) and the exponent will become positive.
As an example:
When simplifying monomials with negative exponents, you can start by ‘flipping over’
all of the negative exponents to make them positive. Then, simplify.
Scan this QR code to go
to a video tutorial on
simplifying monomials
with negative exponents!
Example 9:
Example 10:
6
Exponents Laws of Exponents
Simplify each expression
5.
6.
7.
8.
9.
10.
Polynomials
A.2 The student will perform operations on polynomials, including
b) adding, subtracting, and multiplying polynomials.
Adding and subtracting polynomials is the same as COMBINING LIKE TERMS. In
order for two terms to be like terms, they must have the same variables and the same
exponents.
Like Terms
Each of these terms contain an ‘
therefore they are like terms.
NOT Like Terms
Although these terms have the same
variables, corresponding variables do not
have the same exponents. Therefore,
these are NOT like terms.
‘,
Example 1:
Like terms are underlined here.
Remember that each term
takes the sign in front of it!
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Remember that if you are subtracting a polynomial, you are subtracting all of the terms
(Therefore, you must distribute the negative to each term first!)
Example 2:
Distribute the negative to everything in
the second set of parentheses!
Then, COMBINE LIKE TERMS!
Polynomials
Scan this QR code to go
to a video tutorial on
adding and subtracting
polynomials.
Simplify each expression
1.
2.
3.
4.
5.
To multiply a polynomial by a monomial, simply distribute the monomial to each term in
the polynomial. You will use the rules of exponents to simplify each term.
Example 3:
Distribute the
to each term.
Then, simplify each term
Example 4:
Don’t forget to check for like terms!
Scan this QR code to go
to a video tutorial on
multiplying monomials
and polynomials.
8
To multiply two polynomials together, distribute each term in the first polynomial to
each term in the second polynomial.
When you are multiplying two binomials together this may be called FOIL.
FOIL stands for:
F – First – multiply the first term in each binomial together
O – Outer – multiply the outermost term in each binomial together
I – Inner – multiply the innermost term in each binomial together
L – Last – multiply the last term in each binomial together
(This is the exact same as distributing the first term, then distributing the second term)
Don’t forget to combine like terms when possible.
Example 5:
First
Outer
Inner
Last
Example 6:
Example 7:
Remember that to square something
means to multiply it by itself!
Polynomials
6.
7.
8.
9.
10.
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Factoring
A.2 The student will perform operations on polynomials, including
c) factoring completely first- and second-degree binomials and trinomials in one or two
variables.
The prime factorization of a number or monomial is that number or monomial broken
down into the product of its prime factors.
Example 1: Write the prime factorization of
9
3
2
3
or
To find the greatest common factor (GCF) of two or more monomials, break each down
into its prime factorization. The GCF is the product of all of the shared factors.
Example 2: What is the greatest common factor of
Circle each factor that they ALL have
in common!
You can use the GCF to help you rewrite (factor) polynomials. If all of the terms in the
polynomial have common factors you can pull these factors out from the terms to factor
the polynomial.
Example 3: Factor
GCF =
Pull the GCF out from each term and rewrite. Check your work by distributing.
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Example 4: Factor
GCF =
Pull the GCF out from each term and rewrite. Check your work by distributing.
Scan this QR code to go
to a video tutorial on
greatest common factors.
Factoring
1. Write the prime factorization of
2. Find the greatest common factor of
3. Factor
Simplifying Radicals
A.3 The student will express the square roots and cube roots of whole numbers and
monomial algebraic expressions in simplest radical form.
To simplify a radical, you will pull out any perfect square factors (i.e. 4, 9, 16, 25, etc.)
The square root of 9 is equal to 3, so you can pull the square root of 9 from underneath
the radical sign to find the simplified answer
, which means 3 times the square
root of 2. You can check this simplification in your calculator by verifying that
.
Another way to simplify radicals, if you don’t know the factors of a number, is to create
a factor tree and break the number down to its prime factors. When you have broken
the number down to all of its prime factors you can pull out pairs of factors for square
roots, which will multiply together to make perfect squares.
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Example 5:
Simplify
2
64
8
4
2
Example 6: Simplify
2
2
16 2 x x x
4
2
8
2
4
2
2
y
4
2 2
2
To simplify a root of a higher index, you pull out factors that occur the same number of
times as the index of the radical. As an example, if you are simplifying
, you
would only pull out factors that occurred 5 times, since 5 is the index of the root.
Example 7: Simplify
Because this is a cube root, I pulled out
things that occurred 3 times.
Scan this QR code to go to
a video tutorial on
simplifying radicals.
Simplifying Radicals
Simplify the following radicals.
4.
5.
6.
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Factoring Special Cases
A.2 The student will perform operations on polynomials, including
c) factoring completely first- and second-degree binomials and trinomials in one or two
variables.
A.3 The student will express the square roots and cube roots of whole numbers and the
square root of a monomial algebraic expression in simplest radical form.
Factoring Trinomials
To factor a trinomial of the form
to , and whose product is equal to
, first find two integers whose sum is equal
.
You can start by listing all of the factors of
to the coefficient of .
, and then see which two factors add up
Once you have determined which factors to use, you can put all of your terms
“in a box” and factor the rows and columns.
Example 1: Factor
So, we are looking for factors of 8 that add up to 6!
Factors of 8
1, 8
2, 4
Sum of factors
9
6
Put terms “in a box”
Find the greatest common
factor in each row and each
column. These will give you
your two binomials!
First
One
Term Factor
(
)
Other
Last
Factor Term
(c)
Check your answer by FOIL-ing!
Sometimes you will not be able to find factors of
happens, the trinomial is PRIME.
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that sum to b. When this
Example 2: Factor
So, we are looking for factors of -4 that add up to 5!
Factors of 8
Sum of factors
1, -4
-3
-1, 4
3
-2, 2
0
Nothing works, therefore this trinomial is PRIME
When factoring, anytime the term is negative and the term is positive, your answer
will have two minus signs!
Example 3: Factor
So, we are looking for factors of 80 that add up to
Factors of 80
-4, -20
-5, -16
!
Sum of factors
-24
-21
Find the greatest common
factor in each row and each
column. These will give you
your two binomials!
Check your answer by FOIL-ing!
Example 4: Factor
Pull out a GCF first!!
So, we are looking for factors of 15 that add up to !
Find the greatest common
factor in each row and each
column. These will give you
your two binomials!
Check your answer by FOIL-ing! Don’t forget your GCF in the front.
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Factoring Special Cases
Factor each of the trinomials below
1.
Scan this QR code to go to
a video tutorial on
factoring trinomials.
2.
3.
4.
To solve a quadratic equation (i.e. find its solutions, roots, or zeros), set one side equal
to zero (put the quadratic in standard form), then factor. Set each factor equal to zero
to find the values for that are the solutions to the quadratic.
Example 5: Find the zeros of
Start by getting one side equal to zero and write in standard form.
Now factor the trinomial.
We are looking for factors of
that add up to
.
and
work!
Set both factors equal to zero!
or
You can check your answer in your calculator by graphing the quadratic. The solutions
are the x-intercepts, so this graph should cross the x-axis at -2 and 9.
-2
15
9
Factoring Special Cases
Find the solution to each trinomial
Scan this QR code to go to
a video tutorial on solving
trinomials by factoring.
5.
6.
7.
Special Cases
A perfect square trinomial can be factored to two binomials that are the same, so you
can write it as the binomial squared.
Example 6:
Factor
If your first and last terms are perfect squares you can check for a perfect square
trinomial. Take the square root of the first and last number and see if the product
of those is equal to ½ of the middle number.
Now that we know this case works, you can write the binomial factor squared
Remember to check your answer by FOIL-ing the binomials back out!
Another special case is if the quadratic is represented as the difference of two perfect
squares (i.e.
). If both the first and last term are perfect squares, and the two
terms are being subtracted their factorization can be written as
. As an
example
. Remember that you can check your work by
FOIL-ing.
Example 7: Factor completely
To begin, you should factor out a GCF. In this case it would be 3.
Now you are left with a difference of squares!
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Factoring Special Cases
Scan this QR code to go to a
video tutorial on factoring
special cases.
8.
9.
10.
Solving Quadratic Equations
A.4 The student will solve multistep linear and quadratic equations in two variables
c)
solving quadratic equations algebraically and graphically;
Graphing a quadratic equation
Standard form for a quadratic function is:
The graph of a quadratic equation will be a parabola.
If
, then the parabola opens upward. If
, then the parabola opens downward.
The axis of symmetry is the line
The x-coordinate of the vertex is
.
. The y-coordinate of the vertex is found by
plugging that x value into the equation and solving for
The y-intercept is
To graph a quadratic:
1. Identify a, b, and c.
2. Find the axis of symmetry (
), and lightly sketch.
3. Find the vertex. The x-coordinate is
. Use this to find the y-coordinate.
4. Plot the y-intercept (c), and its reflection across the axis of symmetry.
5. Draw a smooth curve through your points.
The vertex of a parabola is its turning point, or the ‘tip’ of the parabola. In this picture,
the turning point is at (2, 0).
Scan this QR code to go to a
video tutorial on graphing and
solving quadratic equations.
17
Example 1: Graph
Step 1: Identify a, b, and c.
Step 2: Find and sketch the axis of symmetry.
Step 3: Find the vertex.
The x-coordinate is 1. Plug this in to find y.
The vertex is (1, 1).
Step 4: Plot the y-intercept and its reflection.
Because c = 3, the y-intercept is (0, 3). Reflecting this point across x = 1
gives the point (2, 3).
Step 5: Draw a smooth curve.
Remember to check your graphs in your calculator!
You might be asked to find the solutions of a quadratic equation by graphing it. The
solutions to a quadratic equation are the points where it crosses the x-axis.
A quadratic can have two solutions, only one solution, or no solutions at all.
Two Solutions
(the parabola
crosses the x-axis
twice)
One Solution
(the parabola
crosses the x-axis
one time)
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No Solutions
(the parabola does
not cross the xaxis)
Sometimes you will need to find the solution to a quadratic that cannot be factored. In
that case, you can use the quadratic formula:
You just substitute the values for a, b, and c into the quadratic formula and simplify.
Example 2: Solve
Plug these values into the quadratic formula
Your two solutions are
and
Scan this QR code to go to a
video tutorial on using the
quadratic formula.
Solving Quadratic Equations
1. Sketch the graph of
2. Sketch the graph of
3. Find the solution(s) by graphing
4. Find the solution(s) by graphing
5. Find the solution(s), use the quadratic formula
6. Find the zero(s) of the quadratic, use any method you like.
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=0
Answers to the
Laws of Exponents
problems:
Factoring Special Cases
1.
2.
3.
4.
5.
1.
2.
3.
4. Prime
5.
6.
6.
7.
7.
8.
9.
10.
or
or
or
8.
9.
or
10.
Solving Quadratic Equations
Polynomials
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1.
Factoring & Simplifying Radicals
3.
1.
2.
3.
4.
5.
6.
2.
or
4.
or
5.
6. No Solution
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