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Unit 1 Notes
Conversions, Polynomials, and
Solving Equations
Interpreting Terms, Factors,
and Coefficients
Vocabulary
• A variable is a letter or symbol used to
represent the unknown
– Example: x, y, A, V
• A constant is a value that does not change;
can be positive or negative
– Example: 4, -32, pi
• A numerical expression may only contain
constants and/or operations
– Example: 4 + 3, 100 10
• An algebraic expression may contain
variables, constants, and/or operations
– Example: x + 4, x2 - 9
• Factors are items that are being multiplied
together
– Examples: 7(5), 4(x + 2)
• An equation is an expression with an equal
sign (=).
– Example: 4x = 8
• Terms refers to how many items are being
added, subtracted, or divided
– Example: x2 + x - 2 has three terms (x2, x, -2)
• Like terms have the same variables raised
to the same power.
– Example: x2 and 5x2 are like terms
– Example: x2 and x are not like terms
– Example: -3x2 + 4x2 + 6x + 10x + 5 - 4
• A coefficient is the number in front of the
variable.
• The base is the part of the expression that
has been raised to a power.
• The exponent is the number being raised
up next to the base; also described as the
number of times you multiply something by
itself
4x
2
10
2
7x
• The leading coefficient is the coefficient of
the term with the largest degree.
– Example: 3x2 + 5
– Example: 6x – 3 + 12x2
– Example: -2x + 4x3 + 3
Classifying Expressions by Degree
Degree
0
Example
5x0 or 5
Name
Constant
1
7x1 or 7x
Linear
2
10x2 + 7x
Quadratic
3
6x3 - 10x2 + 7x
Cubic
4
8x4 – 5
Quartic
5
3x5 -6x3 + 5
Quintic
>5
7x7
–
4x2
+ 92
Polynomial w/ #
degree
Classifying Expressions by Terms
# of
Terms
Example
Name
1
5, 7x, 2x2
Mononomial
2
3x + 7
Binomial
3
5x2 + 2x – 8
Trinomial
>3
8x4
-
3x2
+ 4x - 9
Polynomial w/ #
terms
Standard Form: putting the terms in order of
highest degree to lowest degree
Example: 9x + 7 – 2x2
1. Rewrite the above expression in standard
form.
2. Identify the coefficients of the expression.
Example: 9x + 7 – 2x2
3. Identify the expression by terms. (How
many terms are there?)
4. Identify the expression by degree. (What is
the highest degree in the expression?)
Example: 2x4y3
1. Identify all the exponents.
2. Identify all the factors.
Simplify the expression by combining like
terms:
8x2 + 12x2 - 9y + 7y + 45 - 10
Properties of Equality
Properties of Operations
Associative Property of Addition: Regardless of the
grouping, the sum of multiple addends will be the same.
(a + b) + c = a + (b + c)
Commutative Property of Addition: Regardless of the
order, the sum of multiple addends will be the same.
a+b=b+a
Additive Identity Property of 0: The sum of any number
and zero is that number.
a+0=a
Additive Inverse: The inverse of a is –a, so that their sum is
zero.
a + (-a) = 0
Properties of Operations
Associative Property of Multiplication: Regardless of the
grouping of the factors, the product will be the same.
(ab)c = a(bc)
Commutative Property of Multiplication: Regardless of the
order of the factors, the product will be the same.
ab = ba
Multiplicative Identity of 1: The product of any number and
1 is that number.
a•1=a
Multiplicative Inverse: The inverse of a is (1/a), so that the
product is one.
a • (1/a) = 1
Properties of Operations
Distributive Property: The sum of two numbers (a and b)
multiplied by a third number (c) is equal to the sum of each
addend (a and b) multiplied by the third number (c).
c(a + b) = ac + bc
Name the property for each of the following:
1. 5 + 6 = 6 + 5
2. 3(5 + 6) = (3 • 5) + (3 • 6)
3. 4 • (7 • 8) = (4 • 7) • 8
4. 78 + 0 = 78
5. 6 • (1/6) = 1
Properties of Equality
Reflexive Property: Anything is equal to itself.
a=a
Symmetric Property: If a = b, then b = a.
Example: 2x = 4 is the same as 4 = 2x
Substitution Property: If a = b, and b equals some quantity,
then it can be substituted for b.
Example: If a = b, and b = 5, then a = 5.
Transitive Property: Using the Substitution Property
If a = b, and b = c, then a = c.
Properties of Equality
Addition Property of Equality: If you add the same number to
both sides of an equation, the equation is true.
If a = b, then a + c = b + c.
Subtraction Property of Equality: If you subtract the same
number to both sides of an equation, the equation is true.
If a = b, then a - c = b - c.
Multiplication Property of Equality: If you multiply the same
number to both sides of an equation, the equation is true.
If a = b, then a • c = b • c.
Division Property of Equality: If you divide the same number to
both sides of an equation, the equation is true.
If a = b, then a/c = b/c.
Name the property for each of the following:
1. If a = b, then a + 5 = b + 5.
2. If a = b, then a/6 = b/5.
3. If a = b, and b = 7, then a = 7.
4. If 5x + 4 = 10, then 10 = 5x + 4.
Solving 1 and 2-step Equations
• An equation is a mathematical statement
that sets two expressions equal to each
other.
– Example: 4x = 8
• The solution to an equation is a value that
makes the equation true.
– Example: If 4x = 8, then x = 2 because…
4x = 8
4(2) = 8
8=8a
How do we find the solution of an equation?
You can rearrange the equation to isolate the
variable by using inverse (opposite)
operations. This is called solving for a
variable.
An equation is like a scale. To make sure
the scale is balanced, perform the same
(inverse) operation to both sides.
4x 8
Given
4x 8
4
4
Divide both sides by 4
(Division Property of
Equality!)
x2
Done!
Solve for x: 5 = 8x + 3
Operations
Properties
Solve for x: 2x + 10 = x - 5
Operations
Properties
Rearranging Formulas
Now, what if we had an equation that only
contained variables – no constants
whatsoever? For example, a + b = c.
What would we do?
Just use inverse (opposite) operations to solve
for the variable specified!
Writing and Solving Equations
Did you know math has it’s own language (e.g
terms, exponents, coefficients)?
What if we didn’t have this language? How
would we get from English phrases back to
numbers and variables?
Each operation (addition, subtraction,
multiplication, and division) has its own key
words that hint at the operation.
Addition
• Sum
• Plus
• Add
• More than (Hint: add
and switch)
• Increased by
• Together
Examples:
• The sum of a number and
7
x+7
• 5 more than a number
x+5
• A number plus 8
x+8
Subtraction
• Difference
• Minus
• Decreased
• Less than (Hint: switch
the order)
• Take Away
Examples:
• The difference of some
number, x, and 7
x–7
• Three less than a
number
x-3
Multiplication
• Product
• Twice
• Double
• Triple
• Times
Examples:
• Product of 5 and 7
5•7
• Twice and Double
2 • x or 2x
• Triple
3 • x or 3x
Division
• Quotient
• Half
• Divide by
Examples:
• The quotient of a and b
a
b
• A number divided by 9
x
9
Exponents
• Square
• Cubed
• To the power of
• Raised to a power
Examples:
• A number squared
x2
• A number to the
power of 5
x5
1. The sum of a number and 10.
2. The product of 9 and x.
3. 7 less than g.
4. The product of 5 and x squared.
5. 9 less than j to the fourth power
Let’s try writing numerical expressions as verbal
translations, or in words.
1. x + 3
2. m – 7
3. 2y
4. k ÷ 5
5. 8 + 3x
6. x5
7. x 7
12
1. Eve reads 25 pages per hour. Write an
expression for the number of pages she
reads in h hours.
2. Sam is 2 years younger than Sue, who is y
years old. Write an expression for Sam’s
age.
3. William runs a mile in 12 minutes. Write
an expression for the number of miles that
William runs in m minutes.
Writing, Solving, and Graphing
Inequalities
What’s an inequality?
• A range of values rather than ONE set
number or answer
• An algebraic relation showing that a
quantity is greater than or less than another
quantity.
Speed limit:
55 x 75
Symbols
Less than
Greater than
Less than OR EQUAL TO
Greater than OR EQUAL TO
Solutions….
You can have a range of answers……
-5 -4 -3 -2 -1 0
1
All real numbers less than 2
x<2
2
3
4
5
Solutions continued…
-5 -4 -3 -2 -1 0
1
2
All real numbers greater than -2
x > -2
3
4
5
Solutions continued….
-5 -4 -3 -2 -1 0
1
2
3
4
All real numbers less than or equal to 1
x 1
5
Solutions continued…
-5 -4 -3 -2 -1 0
1
2
3
4
5
All real numbers greater than or equal to -3
x 3
Did you notice…
Some of the dots were solid
and some were open?
x2
-5 -4 -3 -2 -1
0
1
2
3
4
5
-5 -4 -3 -2 -1
0
1
2
3
4
5
x 1
Why do you think that is?
If the symbol is > or < then dot is open because it can not be
equal.
If the symbol is or then the dot is solid, because it can
be that point too.
Solving an Inequality
Solving a linear inequality in one variable is much like
solving a linear equation in one variable. Isolate the
variable on one side using inverse operations.
Solve using addition:
x–3<5
Add the same number to EACH side.
x 3 5
+3
+3
x<8
Solving Using Subtraction
Subtract the same number from EACH side.
x 6 10
-6
-6
x4
Using Subtraction…
x 5 3
Graph the solution.
-5 -4 -3 -2 -1 0
1
2
3
4
5
Using Addition…
2 n4
Graph the solution.
-5 -4 -3 -2 -1 0
1
2
3
4
5
THE TRAP…..
When you multiply or divide each side of an
inequality by a negative number, you must
reverse the inequality symbol to maintain a
true statement.
Solving using Multiplication
Multiply each side by the same positive number.
1
(2) x 3 (2)
2
x6
Solving Using Division
Divide each side by the same positive number.
3x 9
3
3
x3
Solving by multiplication of a
negative #
Multiply each side by the same negative number and
REVERSE the inequality symbol.
(-1)
x 4 (-1)
Multiply by (-1).
See the switch
x 4
Solving by dividing by a negative
#
Divide each side by the same negative number
and reverse the inequality symbol.
2x 6
-2
-2
x3
Some things to remember!
When you multiply or divide by a negative
number on both sides of the inequality, the
inequality sign changes or "flips."
HINT: When graphing inequalities, always
keep the variable on the LEFT HAND SIDE.
The inequality sign will tell you in which
direction to shade.
Operations with Polynomials
Degree of a Term: the value of the exponent
of the variable
Example: What is the degree of x ?
34
Degree of a Polynomial: the largest degree
of its terms
Example: What is the degree of the
2
7
polynomial 5 x 2 x 8?
Classifying Expressions by Terms
Example: 7 4x x
3
2
Write the following polynomial in standard
form.
State its degree and leading coefficient.
Classify by the degree and number of terms.
Remember!
When combining like terms…
1. Combine the coefficients
2. The exponents do not change!
Simplify.
5 x 2 x 7 x 3x 7 x 5 x
2
2
2
Addition – Vertical Method
When adding two numbers using the vertical
method, we line up the place values vertically:
Hundreds
Tens
Ones
463
+ 239
702
Addition – Vertical Method
Similarly, we put the polynomials in standard
form and line up the like terms of each
polynomial.
Example: 4 x 3 x 2 x 2 8 6 3x 3 x
4x
2x x 8
3
3x
2
3
x3 2 x2
x6
14
Addition – Horizontal Method
When using the horizontal method, identify
the like terms and add them.
Example: 4 x 3 x 2 x 2 8 6 3x 3 x
Subtraction – Vertical Method
To subtract, write the polynomials in
standard form and line up the like terms.
Then, add the opposite.
Example: x 2 3x 5 6 5 x 4 x 2
x 3x 5
x 3x 5
2
4 x 5 x 6
2
2
→
4 x 5 x 6
2
3x 8 x 11
2
Subtraction – Horizontal Method
To subtract, find the like terms and add the
opposite.
Example: x 2 3x 5 6 5 x 4 x 2
Simplify the expression.
2
3
3
2
3
x
x
1
x
2
8
3
x
Simplify the expression.
3x
2
x 1 x 2 8 3x
3
3
2
Multiplying Polynomials
Remember: when multiplying, you are using
the Distributive Property!
So…
1. Multiply the coefficients
2. Add the exponents
3. Combine like terms
Simplify the expression.
5 x 2 x 3x 1
2
Simplify the expression.
5 xy x 3xy 7 y
2
2
Simplify the expression.
2 x y 7 x x 8xy 12 x y
2
2
3
4
3
5
Multiplying Binomials
When multiplying two binomials, you will
have a total of four terms.
}
}
}
}
Example: 2 x 53x 3
1st
2nd
3rd
4th
In order to carry out this operation we must
FOIL (again, the Distributive Property!).
FOIL stands for:
Example: (2x + 5)(3x -3)
First: multiply the first and
third terms in each set of
parentheses
First: 2x and 3x
2x(3x) = 6x2
Outer: multiply the first and
last terms
Outer: 2x and -3
-3(2x) = -6x
Inner: multiply the second
and third terms
Inner: 5 and 3x
5(3x) = 15x
Last: multiply the second and
fourth terms
Last: 5 and -3
-3(5) = -15
Example: (2x + 5)(3x -3)
Outer: 2x and -3
-3(2x) = -6x
Inner: 5 and 3x
5(3x) = 15x
Last: 5 and -3
-3(5) = -15
}
First: 2x and 3x
2x(3x) = 6x2
Next, we combine like terms:
6x2 – 6x + 15x – 15
6x2 + 9x – 15
Done!
Multiply:
2 x
2
3x 1 5 x 7 x 8
2
Dividing Polynomials by Monomials
1. Divide each term in the dividend by the
divisor
2. Simplify each fraction using this rule:
m
x
mn
x
n
x
Divide:
Divide:
Factoring Polynomials
x bx c
2
Find two numbers, m and n, such that:
m+n=b
m•n=c
The solution will be in the form: x mx n
Factor: x 8 x 15
2
Factor: x x 6
2