SECTION 3.1: WHAT IS A THING?
We will define a "thing" as that which can
be "counted," "ordered," or "measured with an instrument."
Some difficult philosophical questions arise when a "thing" is defined in this manner. For example, "Which of our
senses allow us to experience a "thing"? Which leads to the question "What
types of human experiences can be counted or ordered or measured with an
instrument?" Can "love" and "hate" be counted? Can "compassion" be ordered? Can "envy" be measured with an instrument?
SECTION 3.2: WHAT IS A VARIABLE?
What is a variable? Well, we will use 3 different ways to
explain the meaning of the word "variable."

"dictionary" definition
The dictionary definition of variable is any "thing" that changes.

"box model" definition/metaphor
A variable is an empty box that
can have only one number placed into it at a time. The number can be taken out
and a different number can be placed in the box. This process can continue on
indefinitely!

"mathematical" definition
A symbol, usually a letter
of our alphabet but not always, that can be assigned a quantity (i.e. a
number) that can change based on the needs of the mathematical situation at
hand.
ex. 1) x = 4 or x = 7 or x = 12 ex. 2) y = 2 or y = 12 or y = 11
CHARLES BABBAGE
Dec 26 1792  Oct 18 1871
Born Teignmouth, England. Died London, England.
Charles Babbage graduated from Cambridge and at the early age of 24 was elected a fellow of the Royal Society. In 1827 he became Lucasian Professor of Mathematics at Cambridge, a position he held for 12 years although Babbage never taught.
He originated the modern analytic computer. By 1834 he invented the principle of the analytical engine, the forerunner of the modern electronic computer.
In 1830 he published "Reflections on the Decline of Science in England", a controversial work that resulted in the formation, one year later, of the British Association for the Advancement of Science. In 1834 Babbage published his most influential work "On the Economy of Machinery and Manufactures", in which he proposed an early form of operational research.
The computation of logarithms had made him aware of the inaccuracy of human calculation, and he became so obsessed with mechanical computation that he spent pounds 6000 in pursuit of it. A government grant of pounds 17000 was given but support withdrawn in 1842. Although Babbage never built an operational, mechanical computer, his design concepts have been proved correct and recently such a computer has been built following Babbage's own design criteria.
SECTION 3.3: MONOMIAL,
BINOMIALS, TRINOMIALS AND POLYNOMIALS
A monomial is a number, a variable, or a number times one or more
variables. Here are some examples:
ex. 1) 5
ex. 2) x
ex. 3) 5x
ex. 4) 5xyz
ex. 5) 4x^{3}y^{2}zg^{5}
A binomial is two monomials that are added. Here are some
examples:
ex 1) 2 + 3x
ex 2) 6 + 5d
ex 3) 2sdr^{3 } + 3g^{2}d^{5}c
ex 4) 5x  6yg^{2 }can be rewritten as , 5x + 6yg^{2}, hence two monomials that are added
A trinomial is three monomials that are added. Here are some
examples:
ex. 1) 5x^{2 } + 6x + 10
ex. 2) 2xy + 5jh + 12
ex. 3) 2xy  5jh  12
comment: 2xy  5jh  12 can be rewritten as 2xy + 5jh + 12,
hence 3 monomials that are added
A polynomial is a general word for one or more monomials that are
added. Here are some examples:
ex. 1) 2xy
ex. 2) 2x + 7
ex. 3) 5x^{3 } + 2x^{2 } + 1
ex. 4) 3x^{7 } + 3x^{6 } + 2x^{2 } + 9x + 4
IMPORTANT TO REMEMBER
ALL SUBTRACTION DEFAULTS TO ADDITION 
If A and B are real numbers
then A  B = A + B
ex. 1) 5x  7y  3z 12x  14y  30z
= 5x + 7y + 3z + 12x + 14y + 30z
ex. 2) 5  7 + 12 15 = 5 + 7 + 12 + 15 
SECTION 3.4: REVIEW OF THE
WORD "TERM"
A "term" is any math expression that is added.
In the following, 1 + 4 + 6 + 10, 1 is a term, 4 is a term, 6 is a term and 10 is a term.
In the following binomial, 5x + 2, 5x is a term and 2 is a term.
In the following polynomial, 5xy + 2z + 8xr + 10m, 5xy is a term; 2z is a
term;8xr is a term; and 10m is a term
SECTION 3.5: COEFFICIENTS AND
LIKE TERMS
A "coefficient" is a constant that is multiplied by one or
more variables. Here are some examples:
ex. 1) x: the coefficient is understood to be 1
ex. 2) 5x: the
coefficient is 5
ex. 3)5xyz:
the coefficient is 5
ex. 4) 4x^{3 }y^{2 }zg^{5 }:
the coefficient is 4
WHAT ARE LIKE TERMS?

"Like terms" are terms that have the same variables with the same
powers.

Here are some examples OF LIKE TERMS:
ex. 1) 7x, 3x, and 2x are like terms because each term has the same variable
x, and all the x's have the same power, a power of 1.
ex. 2) 3xy^{2 }z, 6xy^{2 }z,
and 6xy^{2 }z are all like terms because each term has the same variables x, y, and z, and
each x has a power of 1, each y has a power of 2, and each z has a power of 1.
ex. 3) 4s^{2 }cz^{3 }, 6s^{2 }cz^{3 }, 40s^{2 }cz^{3} are all like terms because each term has the same variable s, c and z, and each
s has a power of 2, each c has a power of 1, and each z has a power of 3.
The following are NOT LIKE TERMS: 4s^{4}cz^{3 }, 6s^{2}cz^{5}, 40s^{4}cz^{5} Why? Even though the variables are the same in each term the powers of the
variables are NOT THE SAME.
SECTION 3.6: ADDING LIKE TERMS
Before I move on let me clarify what I mean by the the words "noncoefficient part of a term." Here are some examples:
ex. 1) Given 7xyx, xyz is called the noncoefficient part of the term 7xyx.
ex. 2) Give 4sr^{3}u^{6 }, sr^{3}u^{6} is called the noncoefficient part of the term 4sr^{3}u^{6}
From what you can see the noncoefficient part of a term is all the variables following the coefficient of a given term. I'll need you to understand this definition which will be used soon.
GENERAL PRINCIPLE 
LIKE TERMS CAN BE ADDED OR
SUBTRACTED

QUICK REVIEW: Please remember that all subtraction can be rewritten as addition!
ex. 1) 7  2 = 7 + 2 ex. 2) 6  15 = 6 + 15 ex. 3) 10 + 17  34 = 10 + 17 + 34 
THE RULE FOR ADDING LIKE TERMS 
STEP 1) ADDTHE COEFFICIENTS OF THE LIKE TERMS.
STEP 2) TAKE THE RESULT FROM STEP 1 AND MULTIPLY IT BY THE NONCOEFFICIENT PART OF A LIKE TERM.
COMMENT: WHAT IS THE NONCOEFFICIENT PART OF A LIKE TERM?
ex. 1) Given 7xyx; xyz is called the noncoefficient part of the term 7xyx
ex. 2) Given 4x^{2 }y^{3}; x^{2 }y^{3} is called the noncoefficient part of the term 4x^{2 }y^{3} 
 .
Here are some examples of adding like terms:
ex 1) 5x + 4x + 3x +2x
step 1) 5 + 4 + 3 + 2 = 14
step 2) 14 is multiplied by the noncoefficient part of a term which is x.
answer: 14x
ex 2)
4s^{2 }cz^{3 }+
6s^{2 }cz^{3} +
40s^{2}cz^{3}
step 1) 4 + 6 + 40 = 50
step 2) 50 is multiplied by the noncoefficient part of a term which is
s^{2}cz^{3}.
answer:
50s^{2}cz^{3}
ex 3)
300bck^{3}m
+
200bck^{3}m

400bck^{3}m
step 1) 300 + 200  400 = 100
step 2) 100 is multiplied by the noncoefficient part of the terms which is
bck^{ 3}^{ }m
answer:
100bck^{ 3}m

SUMMING UP

So what have you learned in this review is adding and subtracting(adding a negative) like terms. Terms can be added and subtracted just like numbers.



THE SUMMER STUDY GUIDE
BY CHAPTERS

RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE
 CHAPTER 1: THE NUMBER SYSTEM
 CHAPTER 2: ORDER OF OPERATIONS
 CHAPTER 3: VARIABLES, MONOMIALS,
BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS
 CHAPTER 4: SIGNED NUMBERS,
ABSOLUTE VALUE, AND INEQUALITY SYMBOLS
 CHAPTER 5: FACTORS, COMMON
FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS
 CHAPTER 6: PROPERTIES OF NUMBERS
 CHAPTER 7: THE WORLD OF FRACTIONS
 CHAPTER 8: EXPONENTS
 CHAPTER 9: ROOTS
 CHAPTER 10: ALGEBRAIC EXPRESSIONS
 CHAPTER 11: CARTESIAN COORDINATE SYSTEM
 CHAPTER 12: SETS, RELATIONS AND FUNCTIONS
 CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS
 CHAPTER 14: XINTERCEPT(ZERO) AND Y INTERCEPT(B)
 CHAPTER 15: LINES
 CHAPTER 16: FUNCTIONS
 CHAPTER 17: MULTIPLYING POLYNOMIALS
 CHAPTER 18: FACTORING
 CHAPTER 19: RATIONAL EXPRESSIONS
 CHAPTER 20: SOLVING EQUATIONS
 CHAPTER 21:SOLVING INEQUALITIES
 CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS
 CHAPTER 23: QUADRATICS
 CHAPTER 24: CIRCLES
 CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES
 CHAPTER 26: VOLUMES
