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ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 19: RATIONAL EXPRESSIONS

 

2 IMPORTANT DEFINITIONS

termdefinition: any "math expression" that is added or subtracted.

ex. 1) 5 + 4 - 6; 5 is a term and 4 is a factor and 6 is a term.

ex. 2) 2x + 5; 2x is a term and 5 is a term.

ex. 3) -3xy - 7x + 4y + 10 ; -3xy is a term and 7x is a term and 4y is a term and 10 is a term..

 

factordefinition: any "math expression" that is multiplied.

ex. 1) (5)(4) ; 5 is a factor and 4 is a factor.

ex. 2) -5(x + 3) ; -5 is a factor and (x + 3) is a factor.

ex. 3) (3x - 7)(x + 3) ; (3x - 6) is a factor and (x + 3) is a factor.

 

SECTION 19.1: WHAT IS A RATIONAL EXPRESSION?

A "rational expression" is a fraction with variables in the numerator and/or the denominator and the variables have positive integer powers. See the examples below.

ex. 1) $\dfrac{3x+2}{7}$   ex. 2) $\dfrac{3x+5}{4x-6}$   ex. 3) MATH

 

SECTION 19.2: ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

I am assuming that you know how to add, subtract, multiply, and divide numerical fractions which are basic ideas of arithmetic. All the principles you know about fractions in arithmetic apply to algebra.


Figure

ex. 1) MATH

ex. 2) MATH

 


Figure

ex. 1)MATH

 

ex. 2) MATH

 

SECTION 19.3: MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS


Figure

ex. 1) MATH

 

ex. 2) MATH

 

ex. 3) MATH

 

ex. 4) MATH

 

A GENERAL COMMENT ON ADDING, SUBTRACTING,
MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS

Make sure that the answer you give when adding, subtracting, multiplying and dividing is in "LOWEST TERMS." A fraction or rational expression is in lowest terms when the greatest common factor of the numerator and denominator is 1.

 

RULE OF CANCELLING

When simplifying a rational expression(i.e. fraction) you can only cancel common factors. YOU CANNOT CANCEL LIKE TERMS!

Comment: common factors are also called "like factors."

 

THE METHOD FOR REDUCING A RATIONAL
EXPRESSION TO LOWEST TERMS

FACTOR THE NUMERATOR AND DENOMIANTOR, THEN CANCEL THE COMMON FACTORS IN THE NEMERATOR AND DENOMINATOR


ex1
ex2

PLEASE NOTE THAT THE RULE OF CANCELING WAS IN CHAPTER 7.

YOU MAY WANT TO REVIEW THIS.

 

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IMPORTANT HINTS:
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At times the G.C.F. of a numerator and denominator is not clear. Here are some helpful hints:

RULE 1) When presented with all numbers in the numerator and denominator, rewriting the numerator and denominator as a product of primes is a quick way to determine what can cancel in the numerator and denominator.

RULE 2) When presented with polynomials in the numerator and the denominator make sure to check to see if one and or the other can be factored.

Figure

 

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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