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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 3: VARIABLES, MONOMIALS, BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERM

 


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."

  1. "dictionary" definition

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

  2. "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!

    Figure

  3. "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

    CharlesBabbage

    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) 4x3y2zg5


A binomial is two monomials that are added. Here are some examples:

ex 1) 2 + 3x

ex 2) 6 + 5d

ex 3) 2sdr3 + 3g2d5c

ex 4) 5x - 6yg2 can be rewritten as , 5x + -6yg2, hence two monomials that are added


A trinomial is three monomials that are added. Here are some examples:

ex. 1) 5x2 + 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) 5x3 + 2x2 + 1

ex. 4) 3x7 + 3x6 + 2x2 + 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)  -4x3 y2 zg5 : 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) 3xy2 z, 6xy2 z, and  6xy2 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) 4s2 cz3 , 6s2 cz3 , 40s2 cz3 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:
4s4cz3 , 6s2cz5, 40s4cz5 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 "non-coefficient part of a term." Here are some examples:

ex. 1) Given 7xyx, xyz is called the non-coefficient part of the term 7xyx.

ex. 2) Give -4sr3u6 , sr3u6 is called the non-coefficient part of the term -4sr3u6

From what you can see the non-coefficient 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 NON-COEFFICIENT PART OF A LIKE TERM.

COMMENT: WHAT IS THE NON-COEFFICIENT PART OF A LIKE TERM?

ex. 1) Given 7xyx;  xyz is called the non-coefficient part of the term
7xyx

ex. 2) Given -4x2 y3; x2 y3 is called the non-coefficient part of the term -4x2 y3

- .

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 non-coefficient part of a term which is x.

answer: 14x

ex 2) 4s2 cz3 + 6s2 cz3 + 40s2cz3

step 1) 4 + 6 + 40 = 50

step 2) 50 is multiplied by the non-coefficient part of a term which is s2cz3.

answer: 50s2cz3

ex 3) 300bck3m + 200bck3m - 400bck3m

step 1) 300 + 200 - 400 = 100

step 2) 100 is multiplied by the non-coefficient part of the terms which is bck 3 m

answer: 100bck 3m

 

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SUMMING UP
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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.

 

 

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THE SUMMER STUDY GUIDE
BY CHAPTERS

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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: X-INTERCEPT(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