Unit 3: Making Sense of Rational Expressions

Unit 3: Making Sense of Rational Expressions 157

Vocabulary

Use the vocabulary words and definitions below as a reference for this unit.

canceling ......................................... dividing a numerator and a denominator by a common factor to write a fraction in lowest terms or before multiplying fractions Example: 15 24 =

3 • 5 2 • 2 • 2 • 3 =

5 8

1

1

common denominator .................. a common multiple of two or more denominators Example: A common denominator for 14 and 56 is 12.

common factor ............................... a number that is a factor of two or more numbers Example: 2 is a common factor of 6 and 12.

common multiple .......................... a number that is a multiple of two or more numbers Example: 18 is a common multiple of 3, 6, and 9.

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cross multiplication ...................... a method for solving and checking proportions; a method for finding a missing numerator or denominator in equivalent fractions or ratios by making the cross products equal Example: To solve this proportion:

n 9

8 12

12 x n = 9 x 8 12n = 72

n = 7212 n = 6

Solution: 6 9

= 8 12

decimal number ............................ any number written with a decimal point in the number Example: A decimal number falls between two whole numbers, such as 1.5 falls between 1 and 2. Decimal numbers smaller than 1 are sometimes called decimal fractions, such as five-tenths is written 0.5.

denominator ................................... the bottom number of a fraction, indicating the number of equal parts a whole was divided into Example: In the fraction 23 the denominator is 3, meaning the whole was divided into 3 equal parts.

difference ........................................ a number that is the result of subtraction Example: In 16 – 9 = 7, 7 is the difference.

Unit 3: Making Sense of Rational Expressions 159

distributive property .................... the product of a number and the sum or difference of two numbers is equal to the sum or difference of the two products Example: x(a + b) = ax + bx

equation .......................................... a mathematical sentence in which two expressions are connected by an equality symbol Example: 2x = 10

equivalent

(forms of a number) ...................... the same number expressed in different forms Example: 34 , 0.75, and 75%

expression ....................................... a collection of numbers, symbols, and/or operation signs that stands for a number Example: 4r2; 3x + 2y; 25 Expressions do not contain equality (=) or inequality (<, >, ≤, ≥, or ≠) symbols.

factor ................................................ a number or expression that divides evenly into another number; one of the numbers multiplied to get a product Example: 1, 2, 4, 5, 10, and 20 are factors of 20 and (x + 1) is one of the factors of (x2 – 1).

factoring .......................................... expressing a polynomial expression as the product of monomials and polynomials Example: x2 – 5x + 4 = 0

(x – 4)(x – 1) = 0

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fraction ............................................ any part of a whole Example: One-half written in fractional form is 12 .

inequality ....................................... a sentence that states one expression is greater than (>), greater than or equal to (≥), less than (<), less than or equal to (≤), or not equal to (≠) another expression Example: a ≠ 5 or x < 7 or 2y + 3 ≥ 11

integers ........................................... the numbers in the set {… , -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

inverse operation .......................... an action that undoes a previously applied action Example: Subtraction is the inverse operation of addition.

irrational number .......................... a real number that cannot be expressed as a ratio of two integers Example: 2

least common denominator

(LCD) ............................................... the smallest common multiple of the denominators of two or more fractions Example: For 34 and

1 6 , 12 is the least

common denominator.

least common multiple

(LCM) .............................................. the smallest of the common multiples of two or more numbers Example: For 4 and 6, 12 is the least common multiple.

Unit 3: Making Sense of Rational Expressions 161

like terms ........................................ terms that have the same variables and the same corresponding exponents Example: In 5x2 + 3x2 + 6, 5x2 and 3x2 are like terms.

minimum ........................................ the smallest amount or number allowed or possible

multiplicative identity ................. the number one (1); the product of a number and the multiplicative identity is the number itself Example: 5 x 1 = 5

multiplicative property of -1 ...... the product of any number and -1 is the opposite or additive inverse of the number Example: -1(a) = -a and a(-1) = -a

negative numbers ......................... numbers less than zero

numerator ....................................... the top number of a fraction, indicating the number of equal parts being considered Example: In the fraction 23 , the numerator is 2.

order of operations ....................... the order of performing computations in parentheses first, then exponents or powers, followed by multiplication and/or division (as read from left to right), then addition and/or subtraction (as read from left to right); also called algebraic order of operations Example: 5 + (12 – 2) ÷ 2 – 3 x 2 =

5 + 10 ÷ 2 – 3 x 2 = 5 + 5 – 6 =

10 – 6 = 4

Unit 3: Making Sense of Rational Expressions162

polynomial ..................................... a monomial or sum of monomials; any rational expression with no variable in the denominator Examples: x3 + 4x2 – x + 8 5mp2

-7x2y2 + 2x2 + 3

positive numbers .......................... numbers greater than zero

product ............................................ the result of multiplying numbers together Example: In 6 x 8 = 48, 48 is the product.

quotient ........................................... the result of dividing two numbers Example: In 42 ÷ 7 = 6, 6 is the quotient.

ratio .................................................. the comparison of two quantities Example: The ratio of a and b is a:b or ab , where b ≠ 0.

rational expression ....................... a fraction whose numerator and/or denominator are polynomials

Examples: x2 + 1 4x2 + 1

x + 2 5x

8

rational number ............................ a real number that can be expressed as a ratio of two integers

real numbers .................................. the set of all rational and irrational numbers

reciprocals ...................................... two numbers whose product is 1; also called multiplicative inverses

Example: Since 43 = 3 4 x 1, the

reciprocal of 34 is 4 3 .

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simplest form

(of a fraction) ................................. a fraction whose numerator and denominator have no common factor greater than 1 Example: The simplest form of 36 is

1 2 .

simplify an expression ................. to perform as many of the indicated operations as possible

solution ........................................... any value for a variable that makes an equation or inequality a true statement Example: In y = 8 + 9

y = 17 17 is the solution.

substitute ........................................ to replace a variable with a numeral Example: 8(a) + 3

8(5) + 3

sum .................................................. the result of adding numbers together Example: In 6 + 8 = 14, 14 is the sum.

term .................................................. a number, variable, product, or quotient in an expression Example: In the expression 4x2 + 3x + x, 4x2, 3x, and x are terms.

variable ........................................... any symbol, usually a letter, which could represent a number