An _____ is a meaningful combination of numbers, letters, and arithmetic operations.

The numbers in an algebraic expression are called _____ while the letters are called _____.

In "x + 7", x is the _____, and 7 is the _____.

x + 7

x -3y

5(-7)

7/b-3

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

x, y, z, a, b, ∅, ∆

A _____ is a symbol that represents any number from a given replacement set.

The t is the set of values of the variable

A _____ is a symbol that has exactly one number in its replacement set. Any numeral is a _____, such as 7, 4, and 11. Pi (𝜋) is also a _____. If m has a replacement set of {9}, then m is a _____.

equals; is equal to; is

is greater than; is more than

is less than

is not equal to

is greater than or equal to

is less than or equal to

( ) as in 3(4x – y)

[ ] as in 4 – [3 + (x – 5)]

{ } as in x – {5 + [y – (3 + x)]}

_____ as in the fraction

4-7x

_____

y

a statement containing one or more terms connected by plus or minus signs

a number, letter, or symbol whose value is fixed.

the numbers and symbols in a product

a letter used as a factor

either a single number, a letter, or a product of several numbers or letters

the number in an algebraic term

a letter used to represent a number

– a small number written to the right of and slightly above another number or letter to indicate how many times the latter is used as a factor.

a number or letter to which an exponent refers

the product of a number used two or more times as a factor

– algebraic terms that have the same literal factors and in which each letter has the same exponent in all of the terms.

A polynomial with one term is called a _____.

A polynomial with two terms is called a _____.

A polynomial with three terms is called a _____.

Look for plus or minus signs separating the terms. (Although we defined a polynomial as a sum of one or more terms, any one of those could be a negative term.)

Make sure that there are no variables in the denominator of any of the terms, nor any variable under the radical sign (√).

Count the number of terms and name the expressions accordingly.

x^2

x^2 + 2x + 1

2a^2b^3c

-p - 7

We can also classify polynomials using their degree. The _____ in one variable is the number of times that the variables occur as a factor in the monomial.

Term: Degree

-5a^3: 3

The _____ with more than one variable is the total number of times its variable occur as factors.

Term: Degree

-5ab^2c^3:

1 + 2 + 3 = 6

The degree of a polynomial is the _____ of the degrees of its terms.

For example, in the polynomial: a^2b + a^2b^2 - ab:

Term: Degree

a^2b: 2 + 1 = 3

a^2b^2: 2 + 2 =4

ab: 1 + 1 = 2

Thus, the degree of a2b + a2b 2 – ab is 4.

The _____ that has only one variable is the exponent of that variable.

The _____ that has only one variable is the highest power appearing in any of the terms.

The _____ that has more than one variable is the sum of the exponents of the variables.

The _____ in more than one variable is the highest sum of the exponents of the variables in any of the terms.

Combing like terms, such as 3x and 2x, involves an important property of numbers called the _____. The _____ works in two directions:

1. To multiply 3 and (4+2) , apply the Distributive Property of Multiplication over Addition. 3(4+2) = 3 • 4 + 3•2 = 18

2. When used in reverse order, a common factor can be removed from the terms. 3 • 4 + 3 • 2 = 3(4 + 2) = 18

_____ - If two quantities are equal, then one quantity can be replaced by the other. In symbols: For all numbers a and b, if a = b, then a may be replaced by b.

Recall that when dealing with the order of operations, we follow the GEMDAS Rule.

First, simplify expressions within _____.

Recall that when dealing with the order of operations, we follow the GEMDAS Rule.

Then, simplify t

Recall that when dealing with the order of operations, we follow the GEMDAS Rule.

Then simplify _____ and _____ in order from left to right

 It is a mathematical statement that shows two numbers or two expressions are equal.

In the equation x + 4 = 15, the expression x + 4 is called the left-hand side, and 15 is called the right-hand side of the equation.

These equations may involve a polynomial in one variable of degree one. These equations may have one solution, no solution, or an infinite number of solutions. Example: 32 = t − 15 This equation has exactly one solution.

any value of the variable that makes the equation a true statement.

are equations that have the same solution. Example: x – 5 = 6 and x = 11 x = 6 + 5 x = 11

an equation that is satisfied by every number for which both sides are defined.  This type of equation has infinitely many solutions.

Examples:

x + x = 2x

If x = 1, then 1 + 1 = 2(1)

2 = 2

If x = 2, then 2 + 2 = 2(2)

4 = 4

2nd Example:

5(x + 3) = 5x + 15

If x = 1, then 5(1 + 3) = 5(1) + 15

5(4) = 5 + 15

20 = 20

If x = 2, then 5(2 + 3) = 5(2) + 15

5(5) = 10 + 15

25 = 25

an equation that has no solution.

Example:

x = x + 1 If x = 1, then 1 = 1 + 1

1 ≠ 2.

2nd Example:

x – x = 1

0 ≠ 1

∴ No solution

an equation that has at least one solution but is not an identity.

Example:

2x - 4 = 0

2x/2 = 4/2

x = 2

5x – 11 = 29; where X = 8

Is the given number a solution of the equation?

x/9 = 6; where X = 63

Is the given number a solution of the equation?

x – 7 = 5

1/2x = 5

6 - 2x = 2(-x + 3)

5 - x = 0

x - 3 = 4 + x

_____ are rules that allow you to balance, manipulate, and solve equations. A more convenient way of solving equations is through the application of the _____ of equality.

Let x, y, and z be real numbers. If x = y, then x + z = y + z.  Adding the same quantity to both sides of an equation results to equal quantities.

Let x, y, and z be real numbers. If x = y, then x - z = y - z.  Subtracting the same quantity to both sides of an equation results to equal quantities.

Let x, y, and z be real numbers. If x = y, then xz = yz.  Both sides of the equation may be multiplied by equals.

Let x, y, and z be real numbers. If x = y and z ≠ 0, then x/z = y/z.

 Both sides of the equation may be divided by the same nonzero real number

Let x, y, and z be real numbers. If x + y = z and x = y, then y + y = z or x + x = z.  Equals may be substituted for equals.

For any real value of x, y, and z: x = x, y = y, and z = z.  Any number or expression is equal to itself.

For any real value of x, y, and z: If x = y, then y = x.  The expressions on both sides of an equation may be interchanged.

For any real value of x, y, and z: If x = y and y = z, then x = z.  If two quantities are both equal to a third quantity, then they are equal to each other.

For any real value of x, y, and z: For any number x, x + 0 = 0 + x = x  If 0 is added to any number x, then the sum is x.

For any real value of x, y, and z: For any number x, x • 1 = 1 • x = x  If x is multiplied by 1, then the product is x.

For any real value of x, y, and z: For any number x, x • 0 = 0 • x = 0  If 0 is a factor, then the product is zero.

It can be solved by subtracting the same number from each side of the equation so that the variable is isolated on one side of the equation.

 It can be solved by adding the same number on each side of the equation.

It can solve a multiplication equation by dividing.

It can solve a division equation by multiplying.

A _____ is a meaningful collection of things.

An _____ is a thing that belongs to the set. For any two sets A and B, if every element of set A is also an element of set B, then set A is a subset of B.

a process that has several distinct possible outcomes in which the result cannot be predicted with certainty.

any possible result of an experiment

the set of all possible outcomes of an experiment

In both cases, we see that the _____ we want takes place if the outcome of the experiment is an element of the subsets of the sample space. This subset of a sample space is called an _____.

An _____ is a subset of the sample space.

_____ is all outcomes in either or both events. _____, E and F, is denoted by E ∪ F. It is read as “E union F.”

For example: E = {2, 4, 6} and F = {1, 2, 3}

E ∪ F = {1, 2, 3, 4, 6}

_____ is only those events common to both. _____, E and F, is denoted by E ∩ F. It is read as “E intersection F.”

For example: E = {2, 4, 6} and F = {1, 2, 3}

E ∩ F = {2}

The _____ is the set of elements not in E that are in the sample space, S. _____ E is denoted by E’. It is read as “complement of E” or “E prime.”

For example: E’ = {1, 3, 5}

To change from scientific notation with a positive integer exponent to standard notation, move the decimal point to the right, the number of places indicated by the exponent.

7.7 x 10^9 = 7,700,000,000

To write a number expressed in scientific notation with a negative exponent in standard form, move the decimal point to the left, the same number of places as the absolute value of the exponent.

8.6 x 10^-9 = 0.0000000086

All nonzero digits are significant.

 235 has three significant digits: 2, 3, and 5.

 1 487 692 has seven significant digits.

All zeros located between nonzero digits are significant digits.

 205 has three significant digits: 2, 0, and 5.

 4 005 has four significant digits.

Not All digits of the first factor when a number is expressed in scientific notation.

 8.43 × 107 has three significant digits.

 2.050 × 10−3 has four significant digits.

Underscored or specified zeros of a whole number ending in zeros.

 27 000 has two significant digits: 2 and 7.

 27 000 has four significant digits: 2, 7, 0, and 0.

Zeros at the end of a whole number (unless specified to be significant)..

 2 000 has one significant digit: 2.

 3 040 has three significant digits: 3, 0, and 4.

Zeros following the decimal point in a number between 0 and 1.

 0.005 has one significant digit: 5.

 0.0005 has one significant digit: 5.

When approximate quantities are added or subtracted, the result will only have as many decimal places as the quantity with the least number of decimal places.

When approximate quantities are multiplied or divided, the result will only have as many significant figures as the quantity with the least number of significant figures.